id	sid	tid	token	lemma	pos
ejpam-6015	1	1	european	european	PROPN
ejpam-6015	1	2	journal	journal	PROPN
ejpam-6015	1	3	of	of	ADP
ejpam-6015	1	4	pure	pure	ADJ
ejpam-6015	1	5	and	and	CCONJ
ejpam-6015	1	6	applied	applied	ADJ
ejpam-6015	1	7	mathematics	mathematic	NOUN
ejpam-6015	1	8	2025	2025	NUM
ejpam-6015	1	9	,	,	PUNCT
ejpam-6015	1	10	vol	vol	NOUN
ejpam-6015	1	11	.	.	PROPN
ejpam-6015	1	12	18	18	NUM
ejpam-6015	1	13	,	,	PUNCT
ejpam-6015	1	14	issue	issue	NOUN
ejpam-6015	1	15	2	2	NUM
ejpam-6015	1	16	,	,	PUNCT
ejpam-6015	1	17	article	article	NOUN
ejpam-6015	1	18	number	number	NOUN
ejpam-6015	1	19	6015	6015	NUM
ejpam-6015	1	20	issn	issn	VERB
ejpam-6015	1	21	1307	1307	NUM
ejpam-6015	1	22	-	-	SYM
ejpam-6015	1	23	5543	5543	NUM
ejpam-6015	1	24	–	–	PUNCT
ejpam-6015	1	25	ejpam.com	ejpam.com	X
ejpam-6015	1	26	published	publish	VERB
ejpam-6015	1	27	by	by	ADP
ejpam-6015	1	28	new	new	PROPN
ejpam-6015	1	29	york	york	PROPN
ejpam-6015	1	30	business	business	PROPN
ejpam-6015	1	31	global	global	ADJ
ejpam-6015	1	32	on	on	ADP
ejpam-6015	1	33	multivalued	multivalued	ADJ
ejpam-6015	1	34	contractions	contraction	NOUN
ejpam-6015	1	35	via	via	ADP
ejpam-6015	1	36	θ	θ	ADJ
ejpam-6015	1	37	-	-	ADJ
ejpam-6015	1	38	hyperbolic	hyperbolic	ADJ
ejpam-6015	1	39	sine	sine	NOUN
ejpam-6015	1	40	distance	distance	NOUN
ejpam-6015	1	41	functions	function	NOUN
ejpam-6015	1	42	hassen	hassen	PROPN
ejpam-6015	1	43	aydi1,2,∗	aydi1,2,∗	PROPN
ejpam-6015	1	44	,	,	PUNCT
ejpam-6015	1	45	abdelbasset	abdelbasset	NOUN
ejpam-6015	1	46	felhi3	felhi3	PROPN
ejpam-6015	1	47	,	,	PUNCT
ejpam-6015	1	48	irshad	irshad	ADJ
ejpam-6015	1	49	ayoob4	ayoob4	PROPN
ejpam-6015	1	50	,	,	PUNCT
ejpam-6015	1	51	nabil	nabil	NOUN
ejpam-6015	1	52	mlaiki4	mlaiki4	PROPN
ejpam-6015	1	53	1	1	NUM
ejpam-6015	1	54	institut	institut	PROPN
ejpam-6015	1	55	supérieur	supérieur	PROPN
ejpam-6015	1	56	d’informatique	d’informatique	PROPN
ejpam-6015	1	57	et	et	NOUN
ejpam-6015	1	58	des	des	X
ejpam-6015	1	59	techniques	techniques	X
ejpam-6015	1	60	de	de	X
ejpam-6015	1	61	communication	communication	NOUN
ejpam-6015	1	62	,	,	PUNCT
ejpam-6015	1	63	université	université	ADJ
ejpam-6015	1	64	de	de	X
ejpam-6015	1	65	sousse	sousse	PROPN
ejpam-6015	1	66	,	,	PUNCT
ejpam-6015	1	67	h.	h.	PROPN
ejpam-6015	1	68	sousse	sousse	PROPN
ejpam-6015	1	69	4000	4000	NUM
ejpam-6015	1	70	,	,	PUNCT
ejpam-6015	1	71	tunisia	tunisia	PROPN
ejpam-6015	1	72	2	2	NUM
ejpam-6015	1	73	department	department	NOUN
ejpam-6015	1	74	of	of	ADP
ejpam-6015	1	75	mathematics	mathematic	NOUN
ejpam-6015	1	76	and	and	CCONJ
ejpam-6015	1	77	applied	apply	VERB
ejpam-6015	1	78	mathematics	mathematic	NOUN
ejpam-6015	1	79	,	,	PUNCT
ejpam-6015	1	80	sefako	sefako	VERB
ejpam-6015	1	81	makgatho	makgatho	PROPN
ejpam-6015	1	82	health	health	PROPN
ejpam-6015	1	83	sciences	sciences	PROPN
ejpam-6015	1	84	university	university	PROPN
ejpam-6015	1	85	,	,	PUNCT
ejpam-6015	1	86	ga	ga	PROPN
ejpam-6015	1	87	-	-	NOUN
ejpam-6015	1	88	rankuwa	rankuwa	PROPN
ejpam-6015	1	89	,	,	PUNCT
ejpam-6015	1	90	south	south	PROPN
ejpam-6015	1	91	africa	africa	PROPN
ejpam-6015	1	92	3	3	NUM
ejpam-6015	1	93	department	department	PROPN
ejpam-6015	1	94	of	of	ADP
ejpam-6015	1	95	mathematics	mathematics	PROPN
ejpam-6015	1	96	and	and	CCONJ
ejpam-6015	1	97	physics	physics	NOUN
ejpam-6015	1	98	,	,	PUNCT
ejpam-6015	1	99	preparatory	preparatory	PROPN
ejpam-6015	1	100	institute	institute	NOUN
ejpam-6015	1	101	for	for	ADP
ejpam-6015	1	102	engineering	engineering	NOUN
ejpam-6015	1	103	studies	study	NOUN
ejpam-6015	1	104	,	,	PUNCT
ejpam-6015	1	105	carthage	carthage	NOUN
ejpam-6015	1	106	university	university	NOUN
ejpam-6015	1	107	,	,	PUNCT
ejpam-6015	1	108	bizerte	bizerte	NOUN
ejpam-6015	1	109	,	,	PUNCT
ejpam-6015	1	110	tunisia	tunisia	PROPN
ejpam-6015	1	111	4	4	NUM
ejpam-6015	1	112	department	department	NOUN
ejpam-6015	1	113	of	of	ADP
ejpam-6015	1	114	mathematics	mathematic	NOUN
ejpam-6015	1	115	and	and	CCONJ
ejpam-6015	1	116	sciences	science	NOUN
ejpam-6015	1	117	,	,	PUNCT
ejpam-6015	1	118	prince	prince	PROPN
ejpam-6015	1	119	sultan	sultan	PROPN
ejpam-6015	1	120	university	university	PROPN
ejpam-6015	1	121	,	,	PUNCT
ejpam-6015	1	122	riyadh	riyadh	PROPN
ejpam-6015	1	123	11586	11586	NUM
ejpam-6015	1	124	,	,	PUNCT
ejpam-6015	1	125	saudi	saudi	PROPN
ejpam-6015	1	126	arabia	arabia	PROPN
ejpam-6015	1	127	abstract	abstract	NOUN
ejpam-6015	1	128	.	.	PUNCT
ejpam-6015	2	1	very	very	ADV
ejpam-6015	2	2	recently	recently	ADV
ejpam-6015	2	3	,	,	PUNCT
ejpam-6015	2	4	the	the	DET
ejpam-6015	2	5	concept	concept	NOUN
ejpam-6015	2	6	of	of	ADP
ejpam-6015	2	7	θ	θ	ADJ
ejpam-6015	2	8	-	-	ADJ
ejpam-6015	2	9	hyperbolic	hyperbolic	ADJ
ejpam-6015	2	10	sine	sine	NOUN
ejpam-6015	2	11	distance	distance	NOUN
ejpam-6015	2	12	functions	function	NOUN
ejpam-6015	2	13	has	have	AUX
ejpam-6015	2	14	been	be	AUX
ejpam-6015	2	15	introduced	introduce	VERB
ejpam-6015	2	16	by	by	ADP
ejpam-6015	2	17	jleli	jleli	ADJ
ejpam-6015	2	18	and	and	CCONJ
ejpam-6015	2	19	samet	samet	VERB
ejpam-6015	2	20	in	in	ADP
ejpam-6015	2	21	[	[	X
ejpam-6015	2	22	1	1	NUM
ejpam-6015	2	23	]	]	PUNCT
ejpam-6015	2	24	.	.	PUNCT
ejpam-6015	3	1	in	in	ADP
ejpam-6015	3	2	this	this	DET
ejpam-6015	3	3	work	work	NOUN
ejpam-6015	3	4	,	,	PUNCT
ejpam-6015	3	5	we	we	PRON
ejpam-6015	3	6	prove	prove	VERB
ejpam-6015	3	7	some	some	DET
ejpam-6015	3	8	related	related	ADJ
ejpam-6015	3	9	fixed	fix	VERB
ejpam-6015	3	10	points	point	NOUN
ejpam-6015	3	11	results	result	NOUN
ejpam-6015	3	12	for	for	ADP
ejpam-6015	3	13	several	several	ADJ
ejpam-6015	3	14	classes	class	NOUN
ejpam-6015	3	15	of	of	ADP
ejpam-6015	3	16	multivalued	multivalue	VERB
ejpam-6015	3	17	mappings	mapping	NOUN
ejpam-6015	3	18	including	include	VERB
ejpam-6015	3	19	manageable	manageable	ADJ
ejpam-6015	3	20	functions	function	NOUN
ejpam-6015	3	21	on	on	ADP
ejpam-6015	3	22	metric	metric	ADJ
ejpam-6015	3	23	spaces	space	NOUN
ejpam-6015	3	24	.	.	PUNCT
ejpam-6015	4	1	2020	2020	NUM
ejpam-6015	4	2	mathematics	mathematic	NOUN
ejpam-6015	4	3	subject	subject	NOUN
ejpam-6015	4	4	classifications	classification	NOUN
ejpam-6015	4	5	:	:	PUNCT
ejpam-6015	4	6	54e50	54e50	NUM
ejpam-6015	4	7	;	;	PUNCT
ejpam-6015	4	8	54e25	54e25	NUM
ejpam-6015	4	9	;	;	PUNCT
ejpam-6015	4	10	47h10	47h10	NUM
ejpam-6015	4	11	;	;	PUNCT
ejpam-6015	4	12	33b10	33b10	NUM
ejpam-6015	4	13	key	key	ADJ
ejpam-6015	4	14	words	word	NOUN
ejpam-6015	4	15	and	and	CCONJ
ejpam-6015	4	16	phrases	phrase	NOUN
ejpam-6015	4	17	:	:	PUNCT
ejpam-6015	4	18	hyperbolic	hyperbolic	ADJ
ejpam-6015	4	19	function	function	NOUN
ejpam-6015	4	20	,	,	PUNCT
ejpam-6015	4	21	θ	θ	ADJ
ejpam-6015	4	22	-	-	ADJ
ejpam-6015	4	23	hyperbolic	hyperbolic	ADJ
ejpam-6015	4	24	sine	sine	NOUN
ejpam-6015	4	25	distance	distance	NOUN
ejpam-6015	4	26	function	function	NOUN
ejpam-6015	4	27	,	,	PUNCT
ejpam-6015	4	28	multivalued	multivalued	ADJ
ejpam-6015	4	29	mapping	mapping	NOUN
ejpam-6015	4	30	,	,	PUNCT
ejpam-6015	4	31	metric	metric	ADJ
ejpam-6015	4	32	space	space	NOUN
ejpam-6015	4	33	,	,	PUNCT
ejpam-6015	4	34	fixed	fix	VERB
ejpam-6015	4	35	point	point	NOUN
ejpam-6015	4	36	1	1	NUM
ejpam-6015	4	37	.	.	PUNCT
ejpam-6015	4	38	introduction	introduction	NOUN
ejpam-6015	4	39	in	in	ADP
ejpam-6015	4	40	1906	1906	NUM
ejpam-6015	4	41	,	,	PUNCT
ejpam-6015	4	42	fréchet	fréchet	NOUN
ejpam-6015	5	1	[	[	X
ejpam-6015	5	2	2	2	NUM
ejpam-6015	5	3	]	]	PUNCT
ejpam-6015	5	4	defined	define	VERB
ejpam-6015	5	5	the	the	DET
ejpam-6015	5	6	concept	concept	NOUN
ejpam-6015	5	7	of	of	ADP
ejpam-6015	5	8	a	a	DET
ejpam-6015	5	9	metric	metric	ADJ
ejpam-6015	5	10	space	space	NOUN
ejpam-6015	5	11	(	(	PUNCT
ejpam-6015	5	12	ms	ms	NOUN
ejpam-6015	5	13	)	)	PUNCT
ejpam-6015	5	14	.	.	PUNCT
ejpam-6015	6	1	definition	definition	NOUN
ejpam-6015	6	2	1	1	NUM
ejpam-6015	6	3	.	.	PUNCT
ejpam-6015	7	1	[	[	X
ejpam-6015	7	2	2	2	X
ejpam-6015	7	3	]	]	PUNCT
ejpam-6015	7	4	let	let	VERB
ejpam-6015	7	5	x	x	PRON
ejpam-6015	7	6	be	be	AUX
ejpam-6015	7	7	any	any	PRON
ejpam-6015	7	8	nonempty	nonempty	ADV
ejpam-6015	7	9	set	set	VERB
ejpam-6015	7	10	.	.	PUNCT
ejpam-6015	8	1	a	a	DET
ejpam-6015	8	2	function	function	NOUN
ejpam-6015	8	3	d	d	NOUN
ejpam-6015	8	4	:	:	PUNCT
ejpam-6015	8	5	x	x	PROPN
ejpam-6015	8	6	×x	×x	X
ejpam-6015	8	7	→	→	SYM
ejpam-6015	8	8	[	[	X
ejpam-6015	8	9	0,+∞	0,+∞	NUM
ejpam-6015	8	10	)	)	PUNCT
ejpam-6015	8	11	is	be	AUX
ejpam-6015	8	12	said	say	VERB
ejpam-6015	8	13	to	to	PART
ejpam-6015	8	14	be	be	AUX
ejpam-6015	8	15	a	a	DET
ejpam-6015	8	16	distance	distance	NOUN
ejpam-6015	8	17	function	function	NOUN
ejpam-6015	8	18	or	or	CCONJ
ejpam-6015	8	19	metric	metric	ADJ
ejpam-6015	8	20	on	on	ADP
ejpam-6015	8	21	x	x	SYM
ejpam-6015	8	22	if	if	SCONJ
ejpam-6015	8	23	for	for	SCONJ
ejpam-6015	8	24	any	any	DET
ejpam-6015	8	25	ϖ	ϖ	NOUN
ejpam-6015	8	26	,	,	PUNCT
ejpam-6015	8	27	ς	ς	PROPN
ejpam-6015	8	28	,	,	PUNCT
ejpam-6015	8	29	s	s	PART
ejpam-6015	8	30	∈	∈	PROPN
ejpam-6015	8	31	x	x	X
ejpam-6015	8	32	,	,	PUNCT
ejpam-6015	8	33	(	(	PUNCT
ejpam-6015	8	34	i	i	NOUN
ejpam-6015	8	35	)	)	PUNCT
ejpam-6015	8	36	d(ϖ	d(ϖ	PROPN
ejpam-6015	8	37	,	,	PUNCT
ejpam-6015	8	38	ς	ς	NOUN
ejpam-6015	8	39	)	)	PUNCT
ejpam-6015	8	40	=	=	SYM
ejpam-6015	8	41	0	0	NUM
ejpam-6015	8	42	iff	iff	PROPN
ejpam-6015	8	43	ς	ς	PROPN
ejpam-6015	8	44	=	=	SYM
ejpam-6015	8	45	ϖ	ϖ	PROPN
ejpam-6015	8	46	,	,	PUNCT
ejpam-6015	8	47	(	(	PUNCT
ejpam-6015	8	48	ii	ii	NOUN
ejpam-6015	8	49	)	)	PUNCT
ejpam-6015	8	50	d(ϖ	d(ϖ	PROPN
ejpam-6015	8	51	,	,	PUNCT
ejpam-6015	8	52	ς	ς	NOUN
ejpam-6015	8	53	)	)	PUNCT
ejpam-6015	8	54	=	=	NOUN
ejpam-6015	8	55	d(ς,ϖ	d(ς,ϖ	NOUN
ejpam-6015	8	56	)	)	PUNCT
ejpam-6015	8	57	,	,	PUNCT
ejpam-6015	8	58	(	(	PUNCT
ejpam-6015	8	59	iii	iii	NOUN
ejpam-6015	8	60	)	)	PUNCT
ejpam-6015	8	61	d(ϖ	d(ϖ	PROPN
ejpam-6015	8	62	,	,	PUNCT
ejpam-6015	8	63	ς	ς	NOUN
ejpam-6015	8	64	)	)	PUNCT
ejpam-6015	9	1	≤	≤	NOUN
ejpam-6015	9	2	d(ς	d(ς	NOUN
ejpam-6015	9	3	,	,	PUNCT
ejpam-6015	9	4	s	s	PART
ejpam-6015	9	5	)	)	PUNCT
ejpam-6015	9	6	+	+	NUM
ejpam-6015	9	7	d(s,ϖ	d(s,ϖ	NOUN
ejpam-6015	9	8	)	)	PUNCT
ejpam-6015	9	9	(	(	PUNCT
ejpam-6015	9	10	triangle	triangle	NOUN
ejpam-6015	9	11	inequality	inequality	NOUN
ejpam-6015	9	12	)	)	PUNCT
ejpam-6015	9	13	.	.	PUNCT
ejpam-6015	10	1	∗corresponding	∗corresponde	VERB
ejpam-6015	10	2	author	author	NOUN
ejpam-6015	10	3	.	.	PUNCT
ejpam-6015	11	1	doi	doi	NOUN
ejpam-6015	11	2	:	:	PUNCT
ejpam-6015	11	3	https://doi.org/10.29020/nybg.ejpam.v18i2.6015	https://doi.org/10.29020/nybg.ejpam.v18i2.6015	PROPN
ejpam-6015	11	4	email	email	NOUN
ejpam-6015	11	5	addresses	address	NOUN
ejpam-6015	11	6	:	:	PUNCT
ejpam-6015	12	1	hassen.aydi@isima.rnu.tn	hassen.aydi@isima.rnu.tn	INTJ
ejpam-6015	12	2	(	(	PUNCT
ejpam-6015	12	3	h.	h.	PROPN
ejpam-6015	12	4	aydi	aydi	VERB
ejpam-6015	12	5	)	)	PUNCT
ejpam-6015	12	6	,	,	PUNCT
ejpam-6015	12	7	abdelbassetfelhi@gmail.com	abdelbassetfelhi@gmail.com	X
ejpam-6015	12	8	(	(	PUNCT
ejpam-6015	12	9	a.	a.	PROPN
ejpam-6015	12	10	felhi	felhi	PROPN
ejpam-6015	12	11	)	)	PUNCT
ejpam-6015	12	12	,	,	PUNCT
ejpam-6015	12	13	iayoub@psu.edu.sa	iayoub@psu.edu.sa	NOUN
ejpam-6015	12	14	(	(	PUNCT
ejpam-6015	12	15	i.	i.	PROPN
ejpam-6015	12	16	ayoob	ayoob	PROPN
ejpam-6015	12	17	)	)	PUNCT
ejpam-6015	12	18	,	,	PUNCT
ejpam-6015	12	19	nmlaiki@psu.edu.sa	nmlaiki@psu.edu.sa	NOUN
ejpam-6015	12	20	(	(	PUNCT
ejpam-6015	12	21	n.	n.	PROPN
ejpam-6015	12	22	mlaiki	mlaiki	PROPN
ejpam-6015	12	23	)	)	PUNCT
ejpam-6015	12	24	,	,	PUNCT
ejpam-6015	12	25	nmlaiki2012@gmail.com	nmlaiki2012@gmail.com	PUNCT
ejpam-6015	13	1	(	(	PUNCT
ejpam-6015	13	2	n.	n.	PROPN
ejpam-6015	13	3	mlaiki	mlaiki	PROPN
ejpam-6015	13	4	)	)	PUNCT
ejpam-6015	13	5	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6015	14	1	1	1	NUM
ejpam-6015	14	2	copyright	copyright	NOUN
ejpam-6015	14	3	:	:	PUNCT
ejpam-6015	14	4	©	©	PROPN
ejpam-6015	14	5	2025	2025	NUM
ejpam-6015	14	6	the	the	DET
ejpam-6015	14	7	author(s	author(s	NOUN
ejpam-6015	14	8	)	)	PUNCT
ejpam-6015	14	9	.	.	PUNCT
ejpam-6015	15	1	(	(	PUNCT
ejpam-6015	15	2	cc	cc	NOUN
ejpam-6015	15	3	by	by	ADP
ejpam-6015	15	4	-	-	PUNCT
ejpam-6015	15	5	nc	nc	PROPN
ejpam-6015	15	6	4.0	4.0	NUM
ejpam-6015	15	7	)	)	PUNCT
ejpam-6015	15	8	h.	h.	PROPN
ejpam-6015	15	9	aydi	aydi	VERB
ejpam-6015	15	10	et	et	PROPN
ejpam-6015	15	11	al	al	PROPN
ejpam-6015	15	12	.	.	PUNCT
ejpam-6015	15	13	/	/	SYM
ejpam-6015	15	14	eur	eur	PROPN
ejpam-6015	15	15	.	.	PUNCT
ejpam-6015	16	1	j.	j.	PROPN
ejpam-6015	16	2	pure	pure	PROPN
ejpam-6015	16	3	appl	appl	PROPN
ejpam-6015	16	4	.	.	PROPN
ejpam-6015	16	5	math	math	PROPN
ejpam-6015	16	6	,	,	PUNCT
ejpam-6015	16	7	18	18	NUM
ejpam-6015	16	8	(	(	PUNCT
ejpam-6015	16	9	2	2	NUM
ejpam-6015	16	10	)	)	PUNCT
ejpam-6015	16	11	(	(	PUNCT
ejpam-6015	16	12	2025	2025	NUM
ejpam-6015	16	13	)	)	PUNCT
ejpam-6015	16	14	,	,	PUNCT
ejpam-6015	16	15	6015	6015	NUM
ejpam-6015	16	16	2	2	NUM
ejpam-6015	16	17	of	of	ADP
ejpam-6015	16	18	14	14	NUM
ejpam-6015	16	19	banach	banach	ADV
ejpam-6015	16	20	fixed	fix	VERB
ejpam-6015	16	21	point	point	NOUN
ejpam-6015	16	22	(	(	PUNCT
ejpam-6015	16	23	fp	fp	X
ejpam-6015	16	24	)	)	PUNCT
ejpam-6015	16	25	theorem	theorem	NOUN
ejpam-6015	16	26	[	[	X
ejpam-6015	16	27	3	3	NUM
ejpam-6015	16	28	]	]	PUNCT
ejpam-6015	16	29	is	be	AUX
ejpam-6015	16	30	an	an	DET
ejpam-6015	16	31	fundamental	fundamental	ADJ
ejpam-6015	16	32	tool	tool	NOUN
ejpam-6015	16	33	in	in	ADP
ejpam-6015	16	34	the	the	DET
ejpam-6015	16	35	theory	theory	NOUN
ejpam-6015	16	36	of	of	ADP
ejpam-6015	16	37	mss	mss	PROPN
ejpam-6015	16	38	.	.	PUNCT
ejpam-6015	17	1	it	it	PRON
ejpam-6015	17	2	affirms	affirm	VERB
ejpam-6015	17	3	the	the	DET
ejpam-6015	17	4	existence	existence	NOUN
ejpam-6015	17	5	of	of	ADP
ejpam-6015	17	6	a	a	DET
ejpam-6015	17	7	unique	unique	ADJ
ejpam-6015	17	8	fp	fp	NOUN
ejpam-6015	17	9	of	of	ADP
ejpam-6015	17	10	contraction	contraction	NOUN
ejpam-6015	17	11	maps	map	NOUN
ejpam-6015	17	12	on	on	ADP
ejpam-6015	17	13	complete	complete	ADJ
ejpam-6015	17	14	mss	mss	PROPN
ejpam-6015	17	15	.	.	PUNCT
ejpam-6015	18	1	later	later	ADV
ejpam-6015	18	2	,	,	PUNCT
ejpam-6015	18	3	it	it	PRON
ejpam-6015	18	4	has	have	AUX
ejpam-6015	18	5	been	be	AUX
ejpam-6015	18	6	generalized	generalize	VERB
ejpam-6015	18	7	and	and	CCONJ
ejpam-6015	18	8	extended	extend	VERB
ejpam-6015	18	9	in	in	ADP
ejpam-6015	18	10	several	several	ADJ
ejpam-6015	18	11	directions	direction	NOUN
ejpam-6015	18	12	,	,	PUNCT
ejpam-6015	18	13	either	either	CCONJ
ejpam-6015	18	14	by	by	ADP
ejpam-6015	18	15	weakening	weaken	VERB
ejpam-6015	18	16	the	the	DET
ejpam-6015	18	17	topology	topology	NOUN
ejpam-6015	18	18	of	of	ADP
ejpam-6015	18	19	the	the	DET
ejpam-6015	18	20	metric	metric	ADJ
ejpam-6015	18	21	,	,	PUNCT
ejpam-6015	18	22	or	or	CCONJ
ejpam-6015	18	23	by	by	ADP
ejpam-6015	18	24	generalizing	generalize	VERB
ejpam-6015	18	25	the	the	DET
ejpam-6015	18	26	contraction	contraction	NOUN
ejpam-6015	18	27	itself	itself	PRON
ejpam-6015	18	28	.	.	PUNCT
ejpam-6015	19	1	several	several	ADJ
ejpam-6015	19	2	works	work	NOUN
ejpam-6015	19	3	arise	arise	VERB
ejpam-6015	19	4	in	in	ADP
ejpam-6015	19	5	this	this	DET
ejpam-6015	19	6	sense	sense	NOUN
ejpam-6015	19	7	,	,	PUNCT
ejpam-6015	19	8	like	like	ADP
ejpam-6015	19	9	[	[	X
ejpam-6015	19	10	4–9	4–9	X
ejpam-6015	19	11	]	]	X
ejpam-6015	19	12	.	.	PUNCT
ejpam-6015	20	1	on	on	ADP
ejpam-6015	20	2	a	a	DET
ejpam-6015	20	3	ms	ms	NOUN
ejpam-6015	20	4	(	(	PUNCT
ejpam-6015	20	5	x	x	NOUN
ejpam-6015	20	6	,	,	PUNCT
ejpam-6015	20	7	d	d	NOUN
ejpam-6015	20	8	)	)	PUNCT
ejpam-6015	20	9	,	,	PUNCT
ejpam-6015	20	10	cb(x	cb(x	NUM
ejpam-6015	20	11	)	)	PUNCT
ejpam-6015	20	12	is	be	AUX
ejpam-6015	20	13	the	the	DET
ejpam-6015	20	14	set	set	NOUN
ejpam-6015	20	15	of	of	ADP
ejpam-6015	20	16	nonempty	nonempty	ADV
ejpam-6015	20	17	bounded	bound	VERB
ejpam-6015	20	18	and	and	CCONJ
ejpam-6015	20	19	closed	closed	ADJ
ejpam-6015	20	20	subsets	subset	NOUN
ejpam-6015	20	21	of	of	ADP
ejpam-6015	20	22	x.	x.	NOUN
ejpam-6015	20	23	for	for	ADP
ejpam-6015	20	24	π	π	PROPN
ejpam-6015	20	25	,	,	PUNCT
ejpam-6015	20	26	ξ	ξ	PROPN
ejpam-6015	20	27	∈	∈	PROPN
ejpam-6015	20	28	cb(x	cb(x	NUM
ejpam-6015	20	29	)	)	PUNCT
ejpam-6015	20	30	,	,	PUNCT
ejpam-6015	20	31	the	the	DET
ejpam-6015	20	32	hausdorff	hausdorff	NOUN
ejpam-6015	20	33	-	-	PUNCT
ejpam-6015	20	34	pompieu	pompieu	NOUN
ejpam-6015	20	35	metric	metric	NOUN
ejpam-6015	20	36	induced	induce	VERB
ejpam-6015	20	37	by	by	ADP
ejpam-6015	20	38	d	d	PROPN
ejpam-6015	20	39	is	be	AUX
ejpam-6015	20	40	h(π	h(π	PROPN
ejpam-6015	20	41	,	,	PUNCT
ejpam-6015	20	42	ξ	ξ	NOUN
ejpam-6015	20	43	)	)	PUNCT
ejpam-6015	20	44	=	=	SYM
ejpam-6015	20	45	max	max	NOUN
ejpam-6015	20	46	{	{	PUNCT
ejpam-6015	20	47	sup	sup	NOUN
ejpam-6015	20	48	a∈π	a∈π	NOUN
ejpam-6015	20	49	δ(a	δ(a	PROPN
ejpam-6015	20	50	,	,	PUNCT
ejpam-6015	20	51	ξ	ξ	NOUN
ejpam-6015	20	52	)	)	PUNCT
ejpam-6015	20	53	,	,	PUNCT
ejpam-6015	20	54	sup	sup	NOUN
ejpam-6015	20	55	b∈ξ	b∈ξ	NOUN
ejpam-6015	20	56	δ(b	δ(b	PROPN
ejpam-6015	20	57	,	,	PUNCT
ejpam-6015	20	58	π	π	NOUN
ejpam-6015	20	59	)	)	PUNCT
ejpam-6015	20	60	}	}	PUNCT
ejpam-6015	20	61	,	,	PUNCT
ejpam-6015	20	62	where	where	SCONJ
ejpam-6015	20	63	δ(ς	δ(ς	PROPN
ejpam-6015	20	64	,	,	PUNCT
ejpam-6015	20	65	π	π	PROPN
ejpam-6015	20	66	)	)	PUNCT
ejpam-6015	20	67	=	=	SYM
ejpam-6015	20	68	inf{d(ς	inf{d(ς	PROPN
ejpam-6015	20	69	,	,	PUNCT
ejpam-6015	20	70	a	a	PRON
ejpam-6015	20	71	)	)	PUNCT
ejpam-6015	20	72	|	|	ADV
ejpam-6015	20	73	a	a	PRON
ejpam-6015	20	74	∈	∈	PROPN
ejpam-6015	20	75	π	π	NOUN
ejpam-6015	20	76	}	}	PUNCT
ejpam-6015	20	77	is	be	AUX
ejpam-6015	20	78	the	the	DET
ejpam-6015	20	79	distance	distance	NOUN
ejpam-6015	20	80	from	from	ADP
ejpam-6015	20	81	ς	ς	PROPN
ejpam-6015	20	82	to	to	ADP
ejpam-6015	20	83	the	the	DET
ejpam-6015	20	84	set	set	PROPN
ejpam-6015	20	85	π	π	PROPN
ejpam-6015	20	86	.	.	PUNCT
ejpam-6015	20	87	definition	definition	NOUN
ejpam-6015	20	88	2	2	NUM
ejpam-6015	20	89	.	.	PUNCT
ejpam-6015	21	1	let	let	VERB
ejpam-6015	21	2	x	x	PRON
ejpam-6015	21	3	be	be	AUX
ejpam-6015	21	4	any	any	PRON
ejpam-6015	21	5	nonempty	nonempty	ADV
ejpam-6015	21	6	set	set	VERB
ejpam-6015	21	7	.	.	PUNCT
ejpam-6015	22	1	an	an	DET
ejpam-6015	22	2	element	element	NOUN
ejpam-6015	22	3	ς	ς	PROPN
ejpam-6015	22	4	∈	∈	PROPN
ejpam-6015	22	5	x	x	PRON
ejpam-6015	22	6	is	be	AUX
ejpam-6015	22	7	said	say	VERB
ejpam-6015	22	8	to	to	PART
ejpam-6015	22	9	be	be	AUX
ejpam-6015	22	10	a	a	DET
ejpam-6015	22	11	fp	fp	NOUN
ejpam-6015	22	12	of	of	ADP
ejpam-6015	22	13	a	a	DET
ejpam-6015	22	14	multivalued	multivalue	VERB
ejpam-6015	22	15	mapping	mapping	NOUN
ejpam-6015	22	16	t	t	NOUN
ejpam-6015	22	17	:	:	PUNCT
ejpam-6015	22	18	x	x	X
ejpam-6015	22	19	→	→	SYM
ejpam-6015	22	20	2x	2x	NOUN
ejpam-6015	22	21	if	if	SCONJ
ejpam-6015	22	22	ς	ς	PROPN
ejpam-6015	22	23	∈	∈	PROPN
ejpam-6015	22	24	t	t	PROPN
ejpam-6015	22	25	(	(	PUNCT
ejpam-6015	22	26	ς	ς	PROPN
ejpam-6015	22	27	)	)	PUNCT
ejpam-6015	22	28	,	,	PUNCT
ejpam-6015	22	29	where	where	SCONJ
ejpam-6015	22	30	2x	2x	NUM
ejpam-6015	22	31	denotes	denote	VERB
ejpam-6015	22	32	the	the	DET
ejpam-6015	22	33	collection	collection	NOUN
ejpam-6015	22	34	of	of	ADP
ejpam-6015	22	35	all	all	DET
ejpam-6015	22	36	nonempty	nonempty	ADJ
ejpam-6015	22	37	subsets	subset	NOUN
ejpam-6015	22	38	of	of	ADP
ejpam-6015	22	39	x.	x.	PROPN
ejpam-6015	22	40	nadler	nadler	PROPN
ejpam-6015	23	1	[	[	X
ejpam-6015	23	2	10	10	NUM
ejpam-6015	23	3	]	]	PUNCT
ejpam-6015	23	4	studied	study	VERB
ejpam-6015	23	5	the	the	DET
ejpam-6015	23	6	existence	existence	NOUN
ejpam-6015	23	7	of	of	ADP
ejpam-6015	23	8	fps	fps	NOUN
ejpam-6015	23	9	for	for	ADP
ejpam-6015	23	10	multivalued	multivalued	ADJ
ejpam-6015	23	11	contractions	contraction	NOUN
ejpam-6015	23	12	.	.	PUNCT
ejpam-6015	24	1	theorem	theorem	NOUN
ejpam-6015	24	2	1	1	NUM
ejpam-6015	24	3	.	.	PUNCT
ejpam-6015	25	1	[	[	X
ejpam-6015	25	2	10	10	NUM
ejpam-6015	25	3	]	]	X
ejpam-6015	25	4	let	let	VERB
ejpam-6015	25	5	(	(	PUNCT
ejpam-6015	25	6	x	x	NOUN
ejpam-6015	25	7	,	,	PUNCT
ejpam-6015	25	8	d	d	NOUN
ejpam-6015	25	9	)	)	PUNCT
ejpam-6015	25	10	be	be	AUX
ejpam-6015	25	11	a	a	DET
ejpam-6015	25	12	complete	complete	ADJ
ejpam-6015	25	13	ms	ms	NOUN
ejpam-6015	25	14	and	and	CCONJ
ejpam-6015	25	15	t	t	PROPN
ejpam-6015	25	16	:	:	PUNCT
ejpam-6015	25	17	x	x	X
ejpam-6015	25	18	→	→	X
ejpam-6015	25	19	cb(x	cb(x	NUM
ejpam-6015	25	20	)	)	PUNCT
ejpam-6015	25	21	be	be	AUX
ejpam-6015	25	22	a	a	DET
ejpam-6015	25	23	contraction	contraction	NOUN
ejpam-6015	25	24	,	,	PUNCT
ejpam-6015	25	25	i.e.	i.e.	X
ejpam-6015	25	26	,	,	PUNCT
ejpam-6015	25	27	h(tς	h(tς	PROPN
ejpam-6015	25	28	,	,	PUNCT
ejpam-6015	25	29	tϖ	tϖ	NOUN
ejpam-6015	25	30	)	)	PUNCT
ejpam-6015	25	31	≤	≤	NOUN
ejpam-6015	25	32	kd(ϖ	kd(ϖ	VERB
ejpam-6015	25	33	,	,	PUNCT
ejpam-6015	25	34	ς	ς	NOUN
ejpam-6015	25	35	)	)	PUNCT
ejpam-6015	25	36	,	,	PUNCT
ejpam-6015	25	37	for	for	ADP
ejpam-6015	25	38	all	all	DET
ejpam-6015	25	39	ϖ	ϖ	NOUN
ejpam-6015	25	40	,	,	PUNCT
ejpam-6015	25	41	ς	ς	PROPN
ejpam-6015	25	42	∈	∈	PROPN
ejpam-6015	25	43	x	x	NOUN
ejpam-6015	25	44	,	,	PUNCT
ejpam-6015	25	45	where	where	SCONJ
ejpam-6015	25	46	k	k	PROPN
ejpam-6015	25	47	∈	∈	PROPN
ejpam-6015	26	1	[	[	X
ejpam-6015	26	2	0	0	NUM
ejpam-6015	26	3	,	,	PUNCT
ejpam-6015	26	4	1	1	NUM
ejpam-6015	26	5	)	)	PUNCT
ejpam-6015	26	6	.	.	PUNCT
ejpam-6015	27	1	then	then	ADV
ejpam-6015	27	2	,	,	PUNCT
ejpam-6015	27	3	there	there	PRON
ejpam-6015	27	4	is	be	VERB
ejpam-6015	27	5	a	a	DET
ejpam-6015	27	6	fp	fp	NOUN
ejpam-6015	27	7	of	of	ADP
ejpam-6015	27	8	t	t	PROPN
ejpam-6015	27	9	.	.	PUNCT
ejpam-6015	28	1	after	after	ADP
ejpam-6015	28	2	the	the	DET
ejpam-6015	28	3	work	work	NOUN
ejpam-6015	28	4	of	of	ADP
ejpam-6015	28	5	nadler	nadler	NOUN
ejpam-6015	28	6	[	[	X
ejpam-6015	28	7	10	10	NUM
ejpam-6015	28	8	]	]	PUNCT
ejpam-6015	28	9	,	,	PUNCT
ejpam-6015	28	10	many	many	ADJ
ejpam-6015	28	11	fp	fp	NOUN
ejpam-6015	28	12	results	result	NOUN
ejpam-6015	28	13	for	for	ADP
ejpam-6015	28	14	multivalued	multivalued	ADJ
ejpam-6015	28	15	mappings	mapping	NOUN
ejpam-6015	28	16	appeared	appear	VERB
ejpam-6015	28	17	in	in	ADP
ejpam-6015	28	18	literature	literature	NOUN
ejpam-6015	28	19	.	.	PUNCT
ejpam-6015	29	1	for	for	ADP
ejpam-6015	29	2	more	more	ADJ
ejpam-6015	29	3	details	detail	NOUN
ejpam-6015	29	4	,	,	PUNCT
ejpam-6015	29	5	see	see	VERB
ejpam-6015	29	6	[	[	X
ejpam-6015	29	7	11–13	11–13	NUM
ejpam-6015	29	8	]	]	X
ejpam-6015	29	9	.	.	PUNCT
ejpam-6015	30	1	motivated	motivate	VERB
ejpam-6015	30	2	by	by	ADP
ejpam-6015	30	3	the	the	DET
ejpam-6015	30	4	fact	fact	NOUN
ejpam-6015	30	5	that	that	SCONJ
ejpam-6015	30	6	hyperbolic	hyperbolic	ADJ
ejpam-6015	30	7	functions	function	NOUN
ejpam-6015	30	8	have	have	AUX
ejpam-6015	30	9	variant	variant	ADJ
ejpam-6015	30	10	applications	application	NOUN
ejpam-6015	30	11	in	in	ADP
ejpam-6015	30	12	many	many	ADJ
ejpam-6015	30	13	fields	field	NOUN
ejpam-6015	30	14	,	,	PUNCT
ejpam-6015	30	15	like	like	ADP
ejpam-6015	30	16	physics	physics	NOUN
ejpam-6015	30	17	,	,	PUNCT
ejpam-6015	30	18	mathematics	mathematic	NOUN
ejpam-6015	30	19	,	,	PUNCT
ejpam-6015	30	20	engineering	engineering	NOUN
ejpam-6015	30	21	,	,	PUNCT
ejpam-6015	30	22	etc	etc	X
ejpam-6015	30	23	,	,	PUNCT
ejpam-6015	30	24	recently	recently	ADV
ejpam-6015	30	25	,	,	PUNCT
ejpam-6015	30	26	jleli	jleli	ADJ
ejpam-6015	30	27	and	and	CCONJ
ejpam-6015	30	28	samet	samet	VERB
ejpam-6015	31	1	[	[	X
ejpam-6015	31	2	1	1	NUM
ejpam-6015	31	3	]	]	PUNCT
ejpam-6015	31	4	introduced	introduce	VERB
ejpam-6015	31	5	the	the	DET
ejpam-6015	31	6	notion	notion	NOUN
ejpam-6015	31	7	of	of	ADP
ejpam-6015	31	8	θ	θ	ADJ
ejpam-6015	31	9	-	-	ADJ
ejpam-6015	31	10	hyperbolic	hyperbolic	ADJ
ejpam-6015	31	11	sine	sine	NOUN
ejpam-6015	31	12	distance	distance	NOUN
ejpam-6015	31	13	functions	function	NOUN
ejpam-6015	31	14	associated	associate	VERB
ejpam-6015	31	15	to	to	ADP
ejpam-6015	31	16	a	a	DET
ejpam-6015	31	17	certain	certain	ADJ
ejpam-6015	31	18	metric	metric	NOUN
ejpam-6015	31	19	and	and	CCONJ
ejpam-6015	31	20	obtained	obtain	VERB
ejpam-6015	31	21	some	some	DET
ejpam-6015	31	22	nice	nice	ADJ
ejpam-6015	31	23	fps	fps	NOUN
ejpam-6015	31	24	results	result	NOUN
ejpam-6015	31	25	.	.	PUNCT
ejpam-6015	32	1	following	follow	VERB
ejpam-6015	32	2	this	this	DET
ejpam-6015	32	3	direction	direction	NOUN
ejpam-6015	32	4	,	,	PUNCT
ejpam-6015	32	5	we	we	PRON
ejpam-6015	32	6	aim	aim	VERB
ejpam-6015	32	7	to	to	PART
ejpam-6015	32	8	establish	establish	VERB
ejpam-6015	32	9	some	some	DET
ejpam-6015	32	10	fps	fps	NOUN
ejpam-6015	32	11	results	result	NOUN
ejpam-6015	32	12	for	for	ADP
ejpam-6015	32	13	some	some	DET
ejpam-6015	32	14	classes	class	NOUN
ejpam-6015	32	15	of	of	ADP
ejpam-6015	32	16	contractive	contractive	ADJ
ejpam-6015	32	17	multivalued	multivalued	ADJ
ejpam-6015	32	18	mappings	mapping	NOUN
ejpam-6015	32	19	on	on	ADP
ejpam-6015	32	20	mss	mss	PROPN
ejpam-6015	32	21	involving	involve	VERB
ejpam-6015	32	22	the	the	DET
ejpam-6015	32	23	θ	θ	ADJ
ejpam-6015	32	24	-	-	ADJ
ejpam-6015	32	25	hyperbolic	hyperbolic	ADJ
ejpam-6015	32	26	sine	sine	NOUN
ejpam-6015	32	27	distance	distance	NOUN
ejpam-6015	32	28	function	function	NOUN
ejpam-6015	32	29	.	.	PUNCT
ejpam-6015	33	1	for	for	ADP
ejpam-6015	33	2	τ	τ	PROPN
ejpam-6015	33	3	>	>	X
ejpam-6015	33	4	0	0	PROPN
ejpam-6015	33	5	,	,	PUNCT
ejpam-6015	33	6	let	let	VERB
ejpam-6015	33	7	θτ	θτ	PART
ejpam-6015	33	8	be	be	AUX
ejpam-6015	33	9	the	the	DET
ejpam-6015	33	10	collection	collection	NOUN
ejpam-6015	33	11	of	of	ADP
ejpam-6015	33	12	functions	function	NOUN
ejpam-6015	33	13	θ	θ	NOUN
ejpam-6015	33	14	:	:	PUNCT
ejpam-6015	34	1	[	[	X
ejpam-6015	34	2	0,+∞	0,+∞	NUM
ejpam-6015	34	3	)	)	PUNCT
ejpam-6015	34	4	→	→	PUNCT
ejpam-6015	35	1	[	[	X
ejpam-6015	35	2	0,+∞	0,+∞	NUM
ejpam-6015	35	3	)	)	PUNCT
ejpam-6015	35	4	so	so	SCONJ
ejpam-6015	35	5	that	that	SCONJ
ejpam-6015	35	6	θ(t	θ(t	VERB
ejpam-6015	35	7	)	)	PUNCT
ejpam-6015	35	8	≥	≥	PROPN
ejpam-6015	35	9	ctτ	ctτ	NOUN
ejpam-6015	35	10	,	,	PUNCT
ejpam-6015	35	11	(	(	PUNCT
ejpam-6015	35	12	1	1	X
ejpam-6015	35	13	)	)	PUNCT
ejpam-6015	35	14	for	for	ADP
ejpam-6015	35	15	all	all	DET
ejpam-6015	35	16	t	t	PROPN
ejpam-6015	35	17	≥	≥	NOUN
ejpam-6015	35	18	0	0	NUM
ejpam-6015	35	19	,	,	PUNCT
ejpam-6015	35	20	where	where	SCONJ
ejpam-6015	35	21	c	c	PROPN
ejpam-6015	35	22	>	>	X
ejpam-6015	35	23	0	0	PUNCT
ejpam-6015	35	24	is	be	AUX
ejpam-6015	35	25	a	a	DET
ejpam-6015	35	26	constant	constant	ADJ
ejpam-6015	35	27	.	.	PUNCT
ejpam-6015	36	1	definition	definition	NOUN
ejpam-6015	36	2	3	3	X
ejpam-6015	36	3	.	.	PUNCT
ejpam-6015	37	1	let	let	VERB
ejpam-6015	37	2	(	(	PUNCT
ejpam-6015	37	3	x	x	NOUN
ejpam-6015	37	4	,	,	PUNCT
ejpam-6015	37	5	d	d	NOUN
ejpam-6015	37	6	)	)	PUNCT
ejpam-6015	37	7	be	be	AUX
ejpam-6015	37	8	a	a	DET
ejpam-6015	37	9	ms	ms	PROPN
ejpam-6015	37	10	.	.	PROPN
ejpam-6015	37	11	for	for	ADP
ejpam-6015	37	12	all	all	DET
ejpam-6015	37	13	τ	τ	PROPN
ejpam-6015	37	14	>	>	X
ejpam-6015	37	15	0	0	PUNCT
ejpam-6015	37	16	and	and	CCONJ
ejpam-6015	38	1	θ	θ	PROPN
ejpam-6015	38	2	∈	∈	PROPN
ejpam-6015	38	3	θτ	θτ	ADV
ejpam-6015	38	4	,	,	PUNCT
ejpam-6015	38	5	consider	consider	VERB
ejpam-6015	38	6	dθ	dθ	NOUN
ejpam-6015	38	7	:	:	PUNCT
ejpam-6015	38	8	x	x	SYM
ejpam-6015	38	9	2	2	NUM
ejpam-6015	38	10	→	→	SYM
ejpam-6015	38	11	[	[	X
ejpam-6015	38	12	0,+∞	0,+∞	NUM
ejpam-6015	38	13	)	)	PUNCT
ejpam-6015	38	14	as	as	ADP
ejpam-6015	38	15	dθ(ϖ	dθ(ϖ	PROPN
ejpam-6015	38	16	,	,	PUNCT
ejpam-6015	38	17	ς	ς	NOUN
ejpam-6015	38	18	)	)	PUNCT
ejpam-6015	38	19	=	=	SYM
ejpam-6015	38	20	θ(sinh(d(ϖ	θ(sinh(d(ϖ	NOUN
ejpam-6015	38	21	,	,	PUNCT
ejpam-6015	38	22	ς	ς	NOUN
ejpam-6015	38	23	)	)	PUNCT
ejpam-6015	38	24	)	)	PUNCT
ejpam-6015	38	25	)	)	PUNCT
ejpam-6015	38	26	,	,	PUNCT
ejpam-6015	38	27	∀ϖ	∀ϖ	PROPN
ejpam-6015	38	28	,	,	PUNCT
ejpam-6015	38	29	ς	ς	PROPN
ejpam-6015	38	30	∈	∈	PROPN
ejpam-6015	38	31	x	x	NOUN
ejpam-6015	38	32	,	,	PUNCT
ejpam-6015	38	33	where	where	SCONJ
ejpam-6015	38	34	sinh	sinh	VERB
ejpam-6015	38	35	the	the	DET
ejpam-6015	38	36	hyperbolic	hyperbolic	ADJ
ejpam-6015	38	37	sine	sine	NOUN
ejpam-6015	38	38	function	function	NOUN
ejpam-6015	38	39	is	be	AUX
ejpam-6015	38	40	given	give	VERB
ejpam-6015	38	41	as	as	ADP
ejpam-6015	38	42	sinh	sinh	NOUN
ejpam-6015	38	43	t	t	PROPN
ejpam-6015	38	44	=	=	PUNCT
ejpam-6015	38	45	et	et	NOUN
ejpam-6015	38	46	−	−	PROPN
ejpam-6015	38	47	e−t	e−t	NOUN
ejpam-6015	38	48	2	2	NUM
ejpam-6015	38	49	,	,	PUNCT
ejpam-6015	38	50	t	t	PROPN
ejpam-6015	38	51	∈	∈	PROPN
ejpam-6015	38	52	r.	r.	NOUN
ejpam-6015	38	53	the	the	DET
ejpam-6015	38	54	mapping	mapping	NOUN
ejpam-6015	38	55	dθ	dθ	PROPN
ejpam-6015	38	56	is	be	AUX
ejpam-6015	38	57	called	call	VERB
ejpam-6015	38	58	the	the	DET
ejpam-6015	38	59	θ	θ	ADJ
ejpam-6015	38	60	-	-	ADJ
ejpam-6015	38	61	hyperbolic	hyperbolic	ADJ
ejpam-6015	38	62	sine	sine	NOUN
ejpam-6015	38	63	distance	distance	NOUN
ejpam-6015	38	64	function	function	NOUN
ejpam-6015	38	65	associated	associate	VERB
ejpam-6015	38	66	to	to	ADP
ejpam-6015	38	67	the	the	DET
ejpam-6015	38	68	metric	metric	PROPN
ejpam-6015	38	69	d.	d.	PROPN
ejpam-6015	38	70	h.	h.	PROPN
ejpam-6015	38	71	aydi	aydi	VERB
ejpam-6015	38	72	et	et	PROPN
ejpam-6015	38	73	al	al	PROPN
ejpam-6015	38	74	.	.	PUNCT
ejpam-6015	38	75	/	/	SYM
ejpam-6015	38	76	eur	eur	PROPN
ejpam-6015	38	77	.	.	PUNCT
ejpam-6015	39	1	j.	j.	PROPN
ejpam-6015	39	2	pure	pure	PROPN
ejpam-6015	39	3	appl	appl	PROPN
ejpam-6015	39	4	.	.	PROPN
ejpam-6015	39	5	math	math	PROPN
ejpam-6015	39	6	,	,	PUNCT
ejpam-6015	39	7	18	18	NUM
ejpam-6015	39	8	(	(	PUNCT
ejpam-6015	39	9	2	2	NUM
ejpam-6015	39	10	)	)	PUNCT
ejpam-6015	39	11	(	(	PUNCT
ejpam-6015	39	12	2025	2025	NUM
ejpam-6015	39	13	)	)	PUNCT
ejpam-6015	39	14	,	,	PUNCT
ejpam-6015	39	15	6015	6015	NUM
ejpam-6015	39	16	3	3	NUM
ejpam-6015	39	17	of	of	ADP
ejpam-6015	39	18	14	14	NUM
ejpam-6015	39	19	some	some	DET
ejpam-6015	39	20	properties	property	NOUN
ejpam-6015	39	21	of	of	ADP
ejpam-6015	39	22	the	the	DET
ejpam-6015	39	23	θ	θ	ADJ
ejpam-6015	39	24	-	-	ADJ
ejpam-6015	39	25	hyperbolic	hyperbolic	ADJ
ejpam-6015	39	26	sine	sine	NOUN
ejpam-6015	39	27	distance	distance	NOUN
ejpam-6015	39	28	function	function	NOUN
ejpam-6015	39	29	are	be	AUX
ejpam-6015	39	30	provided	provide	VERB
ejpam-6015	39	31	below	below	ADV
ejpam-6015	39	32	.	.	PUNCT
ejpam-6015	40	1	proposition	proposition	NOUN
ejpam-6015	40	2	1	1	NUM
ejpam-6015	40	3	.	.	PUNCT
ejpam-6015	41	1	[	[	X
ejpam-6015	41	2	1	1	X
ejpam-6015	41	3	]	]	X
ejpam-6015	41	4	let	let	VERB
ejpam-6015	41	5	(	(	PUNCT
ejpam-6015	41	6	x	x	NOUN
ejpam-6015	41	7	,	,	PUNCT
ejpam-6015	41	8	d	d	NOUN
ejpam-6015	41	9	)	)	PUNCT
ejpam-6015	41	10	be	be	AUX
ejpam-6015	41	11	a	a	DET
ejpam-6015	41	12	ms	ms	NOUN
ejpam-6015	41	13	and	and	CCONJ
ejpam-6015	41	14	θ	θ	PROPN
ejpam-6015	41	15	∈	∈	PROPN
ejpam-6015	41	16	θτ	θτ	NOUN
ejpam-6015	41	17	for	for	ADP
ejpam-6015	41	18	some	some	PRON
ejpam-6015	41	19	τ	τ	PROPN
ejpam-6015	41	20	>	>	X
ejpam-6015	41	21	0	0	PROPN
ejpam-6015	41	22	.	.	PUNCT
ejpam-6015	42	1	then	then	ADV
ejpam-6015	42	2	,	,	PUNCT
ejpam-6015	42	3	for	for	ADP
ejpam-6015	42	4	all	all	DET
ejpam-6015	42	5	ϖ	ϖ	NOUN
ejpam-6015	42	6	,	,	PUNCT
ejpam-6015	42	7	ς	ς	PROPN
ejpam-6015	42	8	∈	∈	PROPN
ejpam-6015	42	9	x	x	NOUN
ejpam-6015	42	10	,	,	PUNCT
ejpam-6015	42	11	we	we	PRON
ejpam-6015	42	12	have	have	VERB
ejpam-6015	42	13	(	(	PUNCT
ejpam-6015	42	14	i	i	NOUN
ejpam-6015	42	15	)	)	PUNCT
ejpam-6015	42	16	dθ(ϖ	dθ(ϖ	PROPN
ejpam-6015	42	17	,	,	PUNCT
ejpam-6015	42	18	ς	ς	NOUN
ejpam-6015	42	19	)	)	PUNCT
ejpam-6015	42	20	=	=	SYM
ejpam-6015	42	21	0	0	PUNCT
ejpam-6015	43	1	=	=	NOUN
ejpam-6015	43	2	⇒	⇒	X
ejpam-6015	43	3	ς	ς	PROPN
ejpam-6015	43	4	=	=	SYM
ejpam-6015	43	5	ϖ.	ϖ.	PROPN
ejpam-6015	43	6	(	(	PUNCT
ejpam-6015	43	7	ii	ii	NOUN
ejpam-6015	43	8	)	)	PUNCT
ejpam-6015	43	9	if	if	SCONJ
ejpam-6015	43	10	θ(0	θ(0	PROPN
ejpam-6015	43	11	)	)	PUNCT
ejpam-6015	43	12	=	=	SYM
ejpam-6015	43	13	0	0	NUM
ejpam-6015	43	14	,	,	PUNCT
ejpam-6015	43	15	then	then	ADV
ejpam-6015	43	16	dθ(ς	dθ(ς	PUNCT
ejpam-6015	43	17	,	,	PUNCT
ejpam-6015	43	18	ς	ς	PROPN
ejpam-6015	43	19	)	)	PUNCT
ejpam-6015	43	20	=	=	SYM
ejpam-6015	43	21	0	0	X
ejpam-6015	43	22	.	.	PUNCT
ejpam-6015	43	23	(	(	PUNCT
ejpam-6015	43	24	iii	iii	NOUN
ejpam-6015	43	25	)	)	PUNCT
ejpam-6015	43	26	dθ(ϖ	dθ(ϖ	PROPN
ejpam-6015	43	27	,	,	PUNCT
ejpam-6015	43	28	ς	ς	NOUN
ejpam-6015	43	29	)	)	PUNCT
ejpam-6015	43	30	=	=	SYM
ejpam-6015	43	31	dθ(ς,ϖ	dθ(ς,ϖ	NOUN
ejpam-6015	43	32	)	)	PUNCT
ejpam-6015	43	33	.	.	PUNCT
ejpam-6015	44	1	notice	notice	VERB
ejpam-6015	44	2	that	that	SCONJ
ejpam-6015	44	3	a	a	DET
ejpam-6015	44	4	θ	θ	ADJ
ejpam-6015	44	5	-	-	ADJ
ejpam-6015	44	6	hyperbolic	hyperbolic	ADJ
ejpam-6015	44	7	sine	sine	NOUN
ejpam-6015	44	8	distance	distance	NOUN
ejpam-6015	44	9	function	function	NOUN
ejpam-6015	44	10	is	be	AUX
ejpam-6015	44	11	not	not	PART
ejpam-6015	44	12	necessarily	necessarily	ADV
ejpam-6015	44	13	a	a	DET
ejpam-6015	44	14	metric	metric	ADJ
ejpam-6015	44	15	,	,	PUNCT
ejpam-6015	44	16	even	even	ADV
ejpam-6015	44	17	if	if	SCONJ
ejpam-6015	44	18	θ(0	θ(0	PROPN
ejpam-6015	44	19	)	)	PUNCT
ejpam-6015	44	20	=	=	SYM
ejpam-6015	45	1	0	0	X
ejpam-6015	45	2	.	.	PUNCT
ejpam-6015	46	1	the	the	DET
ejpam-6015	46	2	next	next	ADJ
ejpam-6015	46	3	example	example	NOUN
ejpam-6015	46	4	shows	show	VERB
ejpam-6015	46	5	this	this	DET
ejpam-6015	46	6	fact	fact	NOUN
ejpam-6015	46	7	.	.	PUNCT
ejpam-6015	47	1	example	example	NOUN
ejpam-6015	48	1	1	1	NUM
ejpam-6015	48	2	.	.	PUNCT
ejpam-6015	48	3	let	let	VERB
ejpam-6015	48	4	x	x	PUNCT
ejpam-6015	48	5	=	=	SYM
ejpam-6015	48	6	r	r	NOUN
ejpam-6015	48	7	and	and	CCONJ
ejpam-6015	48	8	d(ϖ	d(ϖ	PROPN
ejpam-6015	48	9	,	,	PUNCT
ejpam-6015	48	10	ς	ς	NOUN
ejpam-6015	48	11	)	)	PUNCT
ejpam-6015	48	12	=	=	PRON
ejpam-6015	48	13	|ς	|ς	ADJ
ejpam-6015	48	14	−	−	PROPN
ejpam-6015	49	1	ϖ|	ϖ|	ADV
ejpam-6015	49	2	for	for	ADP
ejpam-6015	49	3	all	all	DET
ejpam-6015	49	4	ϖ	ϖ	NOUN
ejpam-6015	49	5	,	,	PUNCT
ejpam-6015	49	6	ς	ς	PROPN
ejpam-6015	49	7	∈	∈	PROPN
ejpam-6015	49	8	x.	x.	NOUN
ejpam-6015	49	9	let	let	VERB
ejpam-6015	49	10	θ(t	θ(t	PROPN
ejpam-6015	49	11	)	)	PUNCT
ejpam-6015	49	12	=	=	PUNCT
ejpam-6015	49	13	√	√	PROPN
ejpam-6015	49	14	t	t	NOUN
ejpam-6015	49	15	for	for	ADP
ejpam-6015	49	16	all	all	DET
ejpam-6015	49	17	t	t	PROPN
ejpam-6015	49	18	≥	≥	NOUN
ejpam-6015	49	19	0	0	NUM
ejpam-6015	49	20	.	.	PUNCT
ejpam-6015	50	1	then	then	ADV
ejpam-6015	50	2	,	,	PUNCT
ejpam-6015	50	3	θ	θ	PROPN
ejpam-6015	50	4	∈	∈	PROPN
ejpam-6015	50	5	θ	θ	NOUN
ejpam-6015	50	6	1	1	NUM
ejpam-6015	50	7	2	2	NUM
ejpam-6015	50	8	.	.	PUNCT
ejpam-6015	51	1	the	the	DET
ejpam-6015	51	2	θ	θ	ADJ
ejpam-6015	51	3	-	-	ADJ
ejpam-6015	51	4	hyperbolic	hyperbolic	ADJ
ejpam-6015	51	5	sine	sine	NOUN
ejpam-6015	51	6	distance	distance	NOUN
ejpam-6015	51	7	function	function	NOUN
ejpam-6015	51	8	associated	associate	VERB
ejpam-6015	51	9	to	to	ADP
ejpam-6015	51	10	the	the	DET
ejpam-6015	51	11	metric	metric	NOUN
ejpam-6015	51	12	d	d	PROPN
ejpam-6015	51	13	is	be	AUX
ejpam-6015	51	14	defined	define	VERB
ejpam-6015	51	15	by	by	ADP
ejpam-6015	51	16	dθ(ϖ	dθ(ϖ	PROPN
ejpam-6015	51	17	,	,	PUNCT
ejpam-6015	51	18	ς	ς	NOUN
ejpam-6015	51	19	)	)	PUNCT
ejpam-6015	51	20	=	=	SYM
ejpam-6015	51	21	θ(sinh(|ς	θ(sinh(|ς	NOUN
ejpam-6015	51	22	−ϖ|	−ϖ|	NOUN
ejpam-6015	51	23	)	)	PUNCT
ejpam-6015	51	24	)	)	PUNCT
ejpam-6015	51	25	for	for	ADP
ejpam-6015	51	26	all	all	DET
ejpam-6015	51	27	ϖ	ϖ	PROPN
ejpam-6015	51	28	,	,	PUNCT
ejpam-6015	51	29	ς	ς	PROPN
ejpam-6015	51	30	∈	∈	PROPN
ejpam-6015	51	31	x.	x.	NOUN
ejpam-6015	51	32	on	on	ADP
ejpam-6015	51	33	the	the	DET
ejpam-6015	51	34	other	other	ADJ
ejpam-6015	51	35	hand	hand	NOUN
ejpam-6015	51	36	,	,	PUNCT
ejpam-6015	51	37	we	we	PRON
ejpam-6015	51	38	have	have	VERB
ejpam-6015	51	39	dθ(1	dθ(1	NOUN
ejpam-6015	51	40	,	,	PUNCT
ejpam-6015	51	41	5	5	NUM
ejpam-6015	51	42	)	)	PUNCT
ejpam-6015	51	43	dθ(1	dθ(1	NOUN
ejpam-6015	51	44	,	,	PUNCT
ejpam-6015	51	45	3	3	NUM
ejpam-6015	51	46	)	)	PUNCT
ejpam-6015	51	47	+	+	CCONJ
ejpam-6015	52	1	dθ(3	dθ(3	PROPN
ejpam-6015	52	2	,	,	PUNCT
ejpam-6015	52	3	5	5	NUM
ejpam-6015	52	4	)	)	PUNCT
ejpam-6015	52	5	=	=	NOUN
ejpam-6015	52	6	θ(sinh(4	θ(sinh(4	NOUN
ejpam-6015	52	7	)	)	PUNCT
ejpam-6015	52	8	)	)	PUNCT
ejpam-6015	52	9	θ(sinh(2	θ(sinh(2	NOUN
ejpam-6015	52	10	)	)	PUNCT
ejpam-6015	52	11	)	)	PUNCT
ejpam-6015	53	1	+	+	CCONJ
ejpam-6015	53	2	θ(sinh(2	θ(sinh(2	NOUN
ejpam-6015	53	3	)	)	PUNCT
ejpam-6015	53	4	)	)	PUNCT
ejpam-6015	54	1	=	=	SYM
ejpam-6015	54	2	√	√	NUM
ejpam-6015	54	3	sinh(4	sinh(4	NOUN
ejpam-6015	54	4	)	)	PUNCT
ejpam-6015	54	5	2	2	NUM
ejpam-6015	54	6	√	√	NOUN
ejpam-6015	54	7	sinh	sinh	VERB
ejpam-6015	54	8	2	2	NUM
ejpam-6015	54	9	=	=	SYM
ejpam-6015	54	10	1	1	NUM
ejpam-6015	54	11	2	2	NUM
ejpam-6015	54	12	√	√	NOUN
ejpam-6015	54	13	e2	e2	NOUN
ejpam-6015	54	14	+	+	CCONJ
ejpam-6015	55	1	e−2	e−2	PROPN
ejpam-6015	55	2	>	>	SYM
ejpam-6015	55	3	1	1	NUM
ejpam-6015	55	4	,	,	PUNCT
ejpam-6015	55	5	which	which	PRON
ejpam-6015	55	6	shows	show	VERB
ejpam-6015	55	7	that	that	SCONJ
ejpam-6015	55	8	dθ	dθ	PROPN
ejpam-6015	55	9	does	do	AUX
ejpam-6015	55	10	not	not	PART
ejpam-6015	55	11	verify	verify	VERB
ejpam-6015	55	12	the	the	DET
ejpam-6015	55	13	triangle	triangle	NOUN
ejpam-6015	55	14	inequality	inequality	NOUN
ejpam-6015	55	15	.	.	PUNCT
ejpam-6015	56	1	consequently	consequently	ADV
ejpam-6015	56	2	,	,	PUNCT
ejpam-6015	56	3	dθ	dθ	PROPN
ejpam-6015	56	4	is	be	AUX
ejpam-6015	56	5	not	not	PART
ejpam-6015	56	6	a	a	DET
ejpam-6015	56	7	metric	metric	NOUN
ejpam-6015	56	8	on	on	ADP
ejpam-6015	56	9	x.	x.	NOUN
ejpam-6015	56	10	proposition	proposition	NOUN
ejpam-6015	56	11	2	2	NUM
ejpam-6015	56	12	.	.	PUNCT
ejpam-6015	57	1	[	[	X
ejpam-6015	57	2	1	1	X
ejpam-6015	57	3	]	]	X
ejpam-6015	57	4	let	let	VERB
ejpam-6015	57	5	(	(	PUNCT
ejpam-6015	57	6	x	x	NOUN
ejpam-6015	57	7	,	,	PUNCT
ejpam-6015	57	8	d	d	NOUN
ejpam-6015	57	9	)	)	PUNCT
ejpam-6015	57	10	be	be	AUX
ejpam-6015	57	11	a	a	DET
ejpam-6015	57	12	ms	ms	PROPN
ejpam-6015	57	13	.	.	PROPN
ejpam-6015	58	1	(	(	PUNCT
ejpam-6015	58	2	i	i	NOUN
ejpam-6015	58	3	)	)	PUNCT
ejpam-6015	58	4	let	let	VERB
ejpam-6015	58	5	dθ	dθ	PROPN
ejpam-6015	58	6	be	be	AUX
ejpam-6015	58	7	the	the	DET
ejpam-6015	58	8	θ	θ	ADJ
ejpam-6015	58	9	-	-	ADJ
ejpam-6015	58	10	hyperbolic	hyperbolic	ADJ
ejpam-6015	58	11	sine	sine	NOUN
ejpam-6015	58	12	distance	distance	NOUN
ejpam-6015	58	13	function	function	NOUN
ejpam-6015	58	14	associated	associate	VERB
ejpam-6015	58	15	to	to	ADP
ejpam-6015	58	16	d	d	PROPN
ejpam-6015	58	17	,	,	PUNCT
ejpam-6015	58	18	where	where	SCONJ
ejpam-6015	58	19	θ	θ	PROPN
ejpam-6015	58	20	∈	∈	PROPN
ejpam-6015	58	21	θτ	θτ	NOUN
ejpam-6015	58	22	for	for	ADP
ejpam-6015	58	23	some	some	PRON
ejpam-6015	58	24	τ	τ	PROPN
ejpam-6015	58	25	>	>	X
ejpam-6015	58	26	0	0	PROPN
ejpam-6015	58	27	.	.	PUNCT
ejpam-6015	59	1	then	then	ADV
ejpam-6015	59	2	,	,	PUNCT
ejpam-6015	59	3	for	for	ADP
ejpam-6015	59	4	all	all	PRON
ejpam-6015	59	5	ι	ι	X
ejpam-6015	59	6	>	>	X
ejpam-6015	59	7	0	0	NUM
ejpam-6015	59	8	,	,	PUNCT
ejpam-6015	59	9	we	we	PRON
ejpam-6015	59	10	have	have	VERB
ejpam-6015	59	11	ιdθ	ιdθ	NOUN
ejpam-6015	59	12	=	=	PUNCT
ejpam-6015	59	13	dθι	dθι	PROPN
ejpam-6015	59	14	,	,	PUNCT
ejpam-6015	59	15	where	where	SCONJ
ejpam-6015	59	16	θι	θι	NOUN
ejpam-6015	59	17	=	=	SYM
ejpam-6015	59	18	ιθ	ιθ	NOUN
ejpam-6015	59	19	.	.	PUNCT
ejpam-6015	60	1	(	(	PUNCT
ejpam-6015	60	2	ii	ii	NOUN
ejpam-6015	60	3	)	)	PUNCT
ejpam-6015	60	4	let	let	VERB
ejpam-6015	60	5	θ1	θ1	NOUN
ejpam-6015	60	6	,	,	PUNCT
ejpam-6015	60	7	θ2	θ2	PROPN
ejpam-6015	60	8	∈	∈	PROPN
ejpam-6015	60	9	θτ	θτ	NOUN
ejpam-6015	60	10	for	for	ADP
ejpam-6015	60	11	some	some	PRON
ejpam-6015	60	12	τ	τ	PROPN
ejpam-6015	60	13	>	>	X
ejpam-6015	60	14	0	0	PROPN
ejpam-6015	60	15	.	.	PUNCT
ejpam-6015	61	1	then	then	ADV
ejpam-6015	61	2	,	,	PUNCT
ejpam-6015	61	3	dθ1	dθ1	VERB
ejpam-6015	61	4	+	+	CCONJ
ejpam-6015	61	5	dθ2	dθ2	NOUN
ejpam-6015	61	6	=	=	SYM
ejpam-6015	61	7	dθ	dθ	PROPN
ejpam-6015	61	8	,	,	PUNCT
ejpam-6015	61	9	where	where	SCONJ
ejpam-6015	61	10	θ	θ	PROPN
ejpam-6015	61	11	=	=	SYM
ejpam-6015	61	12	θ1	θ1	PROPN
ejpam-6015	61	13	+	+	CCONJ
ejpam-6015	61	14	θ2	θ2	PROPN
ejpam-6015	61	15	.	.	PUNCT
ejpam-6015	62	1	proposition	proposition	NOUN
ejpam-6015	62	2	3	3	NUM
ejpam-6015	62	3	.	.	PUNCT
ejpam-6015	63	1	[	[	X
ejpam-6015	63	2	1	1	X
ejpam-6015	63	3	]	]	PUNCT
ejpam-6015	63	4	let	let	VERB
ejpam-6015	63	5	θ	θ	PROPN
ejpam-6015	63	6	∈	∈	PROPN
ejpam-6015	63	7	θτ	θτ	NOUN
ejpam-6015	63	8	for	for	ADP
ejpam-6015	63	9	some	some	PRON
ejpam-6015	63	10	τ	τ	PROPN
ejpam-6015	63	11	>	>	X
ejpam-6015	63	12	0	0	X
ejpam-6015	63	13	.	.	PUNCT
ejpam-6015	63	14	assume	assume	VERB
ejpam-6015	63	15	that	that	SCONJ
ejpam-6015	63	16	:	:	PUNCT
ejpam-6015	63	17	(	(	PUNCT
ejpam-6015	63	18	i	i	NOUN
ejpam-6015	63	19	)	)	PUNCT
ejpam-6015	63	20	θ(0	θ(0	PROPN
ejpam-6015	63	21	)	)	PUNCT
ejpam-6015	63	22	=	=	PUNCT
ejpam-6015	63	23	0	0	NUM
ejpam-6015	63	24	;	;	PUNCT
ejpam-6015	63	25	(	(	PUNCT
ejpam-6015	63	26	ii	ii	NOUN
ejpam-6015	63	27	)	)	PUNCT
ejpam-6015	63	28	there	there	PRON
ejpam-6015	63	29	exists	exist	VERB
ejpam-6015	63	30	r∗	r∗	PROPN
ejpam-6015	63	31	>	>	X
ejpam-6015	63	32	0	0	NUM
ejpam-6015	64	1	such	such	ADJ
ejpam-6015	64	2	that	that	DET
ejpam-6015	64	3	θ(sinh	θ(sinh	NOUN
ejpam-6015	64	4	r∗	r∗	PROPN
ejpam-6015	64	5	)	)	PUNCT
ejpam-6015	64	6	=	=	NOUN
ejpam-6015	65	1	r∗.	r∗.	NOUN
ejpam-6015	65	2	then	then	ADV
ejpam-6015	65	3	,	,	PUNCT
ejpam-6015	65	4	for	for	ADP
ejpam-6015	65	5	every	every	DET
ejpam-6015	65	6	nonempty	nonempty	ADV
ejpam-6015	65	7	set	set	VERB
ejpam-6015	65	8	x	x	NOUN
ejpam-6015	65	9	,	,	PUNCT
ejpam-6015	65	10	there	there	PRON
ejpam-6015	65	11	exists	exist	VERB
ejpam-6015	65	12	a	a	DET
ejpam-6015	65	13	metric	metric	ADJ
ejpam-6015	65	14	d	d	NOUN
ejpam-6015	65	15	on	on	ADP
ejpam-6015	65	16	x	x	SYM
ejpam-6015	65	17	such	such	ADJ
ejpam-6015	65	18	that	that	SCONJ
ejpam-6015	65	19	the	the	DET
ejpam-6015	65	20	θ	θ	ADJ
ejpam-6015	65	21	-	-	ADJ
ejpam-6015	65	22	hyperbolic	hyperbolic	ADJ
ejpam-6015	65	23	sine	sine	NOUN
ejpam-6015	65	24	distance	distance	NOUN
ejpam-6015	65	25	function	function	NOUN
ejpam-6015	65	26	associated	associate	VERB
ejpam-6015	65	27	to	to	ADP
ejpam-6015	65	28	d	d	PROPN
ejpam-6015	65	29	coincides	coincide	NOUN
ejpam-6015	65	30	with	with	ADP
ejpam-6015	65	31	d	d	PROPN
ejpam-6015	65	32	,	,	PUNCT
ejpam-6015	65	33	i.e.	i.e.	X
ejpam-6015	65	34	,	,	PUNCT
ejpam-6015	65	35	dθ	dθ	PROPN
ejpam-6015	65	36	=	=	PROPN
ejpam-6015	65	37	d.	d.	PROPN
ejpam-6015	65	38	h.	h.	PROPN
ejpam-6015	65	39	aydi	aydi	VERB
ejpam-6015	65	40	et	et	PROPN
ejpam-6015	65	41	al	al	PROPN
ejpam-6015	66	1	.	.	PUNCT
ejpam-6015	66	2	/	/	SYM
ejpam-6015	66	3	eur	eur	PROPN
ejpam-6015	66	4	.	.	PUNCT
ejpam-6015	67	1	j.	j.	PROPN
ejpam-6015	67	2	pure	pure	PROPN
ejpam-6015	67	3	appl	appl	PROPN
ejpam-6015	67	4	.	.	PROPN
ejpam-6015	67	5	math	math	PROPN
ejpam-6015	67	6	,	,	PUNCT
ejpam-6015	67	7	18	18	NUM
ejpam-6015	67	8	(	(	PUNCT
ejpam-6015	67	9	2	2	NUM
ejpam-6015	67	10	)	)	PUNCT
ejpam-6015	67	11	(	(	PUNCT
ejpam-6015	67	12	2025	2025	NUM
ejpam-6015	67	13	)	)	PUNCT
ejpam-6015	67	14	,	,	PUNCT
ejpam-6015	67	15	6015	6015	NUM
ejpam-6015	67	16	4	4	NUM
ejpam-6015	67	17	of	of	ADP
ejpam-6015	67	18	14	14	NUM
ejpam-6015	67	19	2	2	NUM
ejpam-6015	67	20	.	.	PUNCT
ejpam-6015	68	1	the	the	DET
ejpam-6015	68	2	hausdorff	hausdorff	PROPN
ejpam-6015	68	3	θ	θ	PROPN
ejpam-6015	68	4	-	-	ADJ
ejpam-6015	68	5	hyperbolic	hyperbolic	ADJ
ejpam-6015	68	6	sine	sine	NOUN
ejpam-6015	68	7	distance	distance	NOUN
ejpam-6015	68	8	function	function	VERB
ejpam-6015	68	9	our	our	PRON
ejpam-6015	68	10	work	work	NOUN
ejpam-6015	68	11	is	be	AUX
ejpam-6015	68	12	concerned	concern	VERB
ejpam-6015	68	13	with	with	ADP
ejpam-6015	68	14	multivalued	multivalued	ADJ
ejpam-6015	68	15	mappings	mapping	NOUN
ejpam-6015	68	16	t	t	NOUN
ejpam-6015	68	17	:	:	PUNCT
ejpam-6015	68	18	x	x	X
ejpam-6015	68	19	→	→	SYM
ejpam-6015	68	20	2x	2x	NUM
ejpam-6015	68	21	.	.	PUNCT
ejpam-6015	69	1	for	for	ADP
ejpam-6015	69	2	this	this	PRON
ejpam-6015	69	3	,	,	PUNCT
ejpam-6015	69	4	let	let	VERB
ejpam-6015	69	5	(	(	PUNCT
ejpam-6015	69	6	x	x	NOUN
ejpam-6015	69	7	,	,	PUNCT
ejpam-6015	69	8	d	d	NOUN
ejpam-6015	69	9	)	)	PUNCT
ejpam-6015	69	10	be	be	AUX
ejpam-6015	69	11	a	a	DET
ejpam-6015	69	12	ms	ms	PROPN
ejpam-6015	69	13	.	.	PROPN
ejpam-6015	69	14	let	let	VERB
ejpam-6015	69	15	θ	θ	PROPN
ejpam-6015	69	16	∈	∈	PROPN
ejpam-6015	69	17	θτ	θτ	NOUN
ejpam-6015	69	18	for	for	ADP
ejpam-6015	69	19	some	some	DET
ejpam-6015	69	20	τ	τ	PROPN
ejpam-6015	69	21	>	>	X
ejpam-6015	69	22	0	0	PROPN
ejpam-6015	69	23	.	.	PUNCT
ejpam-6015	70	1	for	for	ADP
ejpam-6015	70	2	two	two	NUM
ejpam-6015	70	3	bounded	bounded	ADJ
ejpam-6015	70	4	and	and	CCONJ
ejpam-6015	70	5	closed	closed	ADJ
ejpam-6015	70	6	subsets	subset	NOUN
ejpam-6015	70	7	π	π	PROPN
ejpam-6015	70	8	,	,	PUNCT
ejpam-6015	70	9	ξ	ξ	PROPN
ejpam-6015	70	10	in	in	ADP
ejpam-6015	70	11	x	x	X
ejpam-6015	70	12	,	,	PUNCT
ejpam-6015	70	13	consider	consider	VERB
ejpam-6015	70	14	hθ(π	hθ(π	NOUN
ejpam-6015	70	15	,	,	PUNCT
ejpam-6015	70	16	ξ	ξ	NOUN
ejpam-6015	70	17	)	)	PUNCT
ejpam-6015	70	18	=	=	SYM
ejpam-6015	70	19	max{∆θ(π	max{∆θ(π	PROPN
ejpam-6015	70	20	,	,	PUNCT
ejpam-6015	70	21	ξ),∆θ(ξ	ξ),∆θ(ξ	PROPN
ejpam-6015	70	22	,	,	PUNCT
ejpam-6015	70	23	π	π	NOUN
ejpam-6015	70	24	)	)	PUNCT
ejpam-6015	70	25	}	}	PUNCT
ejpam-6015	70	26	,	,	PUNCT
ejpam-6015	70	27	where	where	SCONJ
ejpam-6015	70	28	∆θ(π	∆θ(π	PROPN
ejpam-6015	70	29	,	,	PUNCT
ejpam-6015	70	30	ξ	ξ	NOUN
ejpam-6015	70	31	)	)	PUNCT
ejpam-6015	70	32	=	=	SYM
ejpam-6015	70	33	sup{θ(sinh(δ(a	sup{θ(sinh(δ(a	PROPN
ejpam-6015	70	34	,	,	PUNCT
ejpam-6015	70	35	ξ	ξ	NOUN
ejpam-6015	70	36	)	)	PUNCT
ejpam-6015	70	37	)	)	PUNCT
ejpam-6015	70	38	)	)	PUNCT
ejpam-6015	70	39	:	:	PUNCT
ejpam-6015	70	40	a	a	DET
ejpam-6015	70	41	∈	∈	PROPN
ejpam-6015	70	42	π	π	NOUN
ejpam-6015	70	43	}	}	PUNCT
ejpam-6015	70	44	=	=	SYM
ejpam-6015	70	45	sup	sup	NUM
ejpam-6015	70	46	a∈π	a∈π	PROPN
ejpam-6015	70	47	inf	inf	PROPN
ejpam-6015	70	48	b∈ξ	b∈ξ	NOUN
ejpam-6015	70	49	{	{	PUNCT
ejpam-6015	70	50	θ(sinh(d(a	θ(sinh(d(a	PROPN
ejpam-6015	70	51	,	,	PUNCT
ejpam-6015	70	52	b	b	NOUN
ejpam-6015	70	53	)	)	PUNCT
ejpam-6015	70	54	)	)	PUNCT
ejpam-6015	70	55	)	)	PUNCT
ejpam-6015	70	56	}	}	PUNCT
ejpam-6015	70	57	.	.	PUNCT
ejpam-6015	71	1	the	the	DET
ejpam-6015	71	2	mapping	mapping	NOUN
ejpam-6015	71	3	hθ	hθ	NOUN
ejpam-6015	71	4	is	be	AUX
ejpam-6015	71	5	called	call	VERB
ejpam-6015	71	6	the	the	DET
ejpam-6015	71	7	hausdorff	hausdorff	PROPN
ejpam-6015	71	8	θ	θ	PROPN
ejpam-6015	71	9	-	-	ADJ
ejpam-6015	71	10	hyperbolic	hyperbolic	ADJ
ejpam-6015	71	11	sine	sine	NOUN
ejpam-6015	71	12	distance	distance	NOUN
ejpam-6015	71	13	function	function	NOUN
ejpam-6015	71	14	associated	associate	VERB
ejpam-6015	71	15	to	to	ADP
ejpam-6015	71	16	the	the	DET
ejpam-6015	71	17	metric	metric	PROPN
ejpam-6015	71	18	d.	d.	PROPN
ejpam-6015	71	19	some	some	DET
ejpam-6015	71	20	properties	property	NOUN
ejpam-6015	71	21	of	of	ADP
ejpam-6015	71	22	hθ	hθ	PROPN
ejpam-6015	71	23	are	be	AUX
ejpam-6015	71	24	provided	provide	VERB
ejpam-6015	71	25	below	below	ADV
ejpam-6015	71	26	.	.	PUNCT
ejpam-6015	72	1	proposition	proposition	NOUN
ejpam-6015	72	2	4	4	NUM
ejpam-6015	72	3	.	.	PUNCT
ejpam-6015	73	1	let	let	VERB
ejpam-6015	73	2	(	(	PUNCT
ejpam-6015	73	3	x	x	NOUN
ejpam-6015	73	4	,	,	PUNCT
ejpam-6015	73	5	d	d	NOUN
ejpam-6015	73	6	)	)	PUNCT
ejpam-6015	73	7	be	be	AUX
ejpam-6015	73	8	a	a	DET
ejpam-6015	73	9	ms	ms	NOUN
ejpam-6015	73	10	and	and	CCONJ
ejpam-6015	73	11	θ	θ	PROPN
ejpam-6015	73	12	∈	∈	PROPN
ejpam-6015	73	13	θτ	θτ	NOUN
ejpam-6015	73	14	for	for	ADP
ejpam-6015	73	15	some	some	PRON
ejpam-6015	73	16	τ	τ	PROPN
ejpam-6015	73	17	>	>	X
ejpam-6015	73	18	0	0	PROPN
ejpam-6015	73	19	.	.	PUNCT
ejpam-6015	74	1	then	then	ADV
ejpam-6015	74	2	,	,	PUNCT
ejpam-6015	74	3	for	for	ADP
ejpam-6015	74	4	every	every	DET
ejpam-6015	74	5	π	π	PROPN
ejpam-6015	74	6	,	,	PUNCT
ejpam-6015	74	7	ξ	ξ	PROPN
ejpam-6015	74	8	∈	∈	PROPN
ejpam-6015	74	9	cb(x	cb(x	NUM
ejpam-6015	74	10	)	)	PUNCT
ejpam-6015	74	11	,	,	PUNCT
ejpam-6015	74	12	(	(	PUNCT
ejpam-6015	74	13	i	i	NOUN
ejpam-6015	74	14	)	)	PUNCT
ejpam-6015	74	15	hθ(π	hθ(π	NOUN
ejpam-6015	74	16	,	,	PUNCT
ejpam-6015	74	17	ξ	ξ	X
ejpam-6015	74	18	)	)	PUNCT
ejpam-6015	74	19	=	=	SYM
ejpam-6015	74	20	0	0	PUNCT
ejpam-6015	74	21	=	=	NOUN
ejpam-6015	74	22	⇒	⇒	X
ejpam-6015	74	23	π	π	X
ejpam-6015	74	24	=	=	SYM
ejpam-6015	74	25	ξ	ξ	PROPN
ejpam-6015	74	26	.	.	PUNCT
ejpam-6015	74	27	(	(	PUNCT
ejpam-6015	74	28	ii	ii	NOUN
ejpam-6015	74	29	)	)	PUNCT
ejpam-6015	74	30	if	if	SCONJ
ejpam-6015	74	31	θ(0	θ(0	PROPN
ejpam-6015	74	32	)	)	PUNCT
ejpam-6015	74	33	=	=	SYM
ejpam-6015	74	34	0	0	NUM
ejpam-6015	74	35	,	,	PUNCT
ejpam-6015	74	36	then	then	ADV
ejpam-6015	74	37	hθ(π	hθ(π	NUM
ejpam-6015	74	38	,	,	PUNCT
ejpam-6015	74	39	π	π	X
ejpam-6015	74	40	)	)	PUNCT
ejpam-6015	74	41	=	=	SYM
ejpam-6015	74	42	0	0	X
ejpam-6015	74	43	.	.	PUNCT
ejpam-6015	74	44	(	(	PUNCT
ejpam-6015	74	45	iii	iii	NOUN
ejpam-6015	74	46	)	)	PUNCT
ejpam-6015	74	47	hθ(π	hθ(π	NOUN
ejpam-6015	74	48	,	,	PUNCT
ejpam-6015	74	49	ξ	ξ	X
ejpam-6015	74	50	)	)	PUNCT
ejpam-6015	74	51	=	=	SYM
ejpam-6015	74	52	hθ(ξ	hθ(ξ	NUM
ejpam-6015	74	53	,	,	PUNCT
ejpam-6015	74	54	π	π	NOUN
ejpam-6015	74	55	)	)	PUNCT
ejpam-6015	74	56	.	.	PUNCT
ejpam-6015	75	1	proof	proof	NOUN
ejpam-6015	75	2	.	.	PUNCT
ejpam-6015	76	1	(	(	PUNCT
ejpam-6015	76	2	i)hθ(π	i)hθ(π	ADP
ejpam-6015	76	3	,	,	PUNCT
ejpam-6015	76	4	ξ	ξ	X
ejpam-6015	76	5	)	)	PUNCT
ejpam-6015	76	6	=	=	SYM
ejpam-6015	76	7	0	0	PUNCT
ejpam-6015	77	1	=	=	NOUN
ejpam-6015	77	2	⇒	⇒	PROPN
ejpam-6015	77	3	∆θ(π	∆θ(π	PROPN
ejpam-6015	77	4	,	,	PUNCT
ejpam-6015	77	5	ξ	ξ	X
ejpam-6015	77	6	)	)	PUNCT
ejpam-6015	77	7	=	=	SYM
ejpam-6015	77	8	∆θ(ξ	∆θ(ξ	PROPN
ejpam-6015	77	9	,	,	PUNCT
ejpam-6015	77	10	π	π	X
ejpam-6015	77	11	)	)	PUNCT
ejpam-6015	77	12	=	=	SYM
ejpam-6015	77	13	0	0	X
ejpam-6015	77	14	.	.	PUNCT
ejpam-6015	78	1	in	in	ADP
ejpam-6015	78	2	the	the	DET
ejpam-6015	78	3	case	case	NOUN
ejpam-6015	78	4	∆θ(π	∆θ(π	PROPN
ejpam-6015	78	5	,	,	PUNCT
ejpam-6015	78	6	ξ	ξ	X
ejpam-6015	78	7	)	)	PUNCT
ejpam-6015	78	8	=	=	SYM
ejpam-6015	78	9	0	0	NUM
ejpam-6015	78	10	,	,	PUNCT
ejpam-6015	78	11	we	we	PRON
ejpam-6015	78	12	get	get	VERB
ejpam-6015	78	13	sup	sup	NOUN
ejpam-6015	78	14	a∈π	a∈π	NOUN
ejpam-6015	78	15	{	{	PUNCT
ejpam-6015	78	16	θ(sinh(δ(a	θ(sinh(δ(a	NOUN
ejpam-6015	78	17	,	,	PUNCT
ejpam-6015	78	18	ξ	ξ	NOUN
ejpam-6015	78	19	)	)	PUNCT
ejpam-6015	78	20	)	)	PUNCT
ejpam-6015	78	21	)	)	PUNCT
ejpam-6015	78	22	}	}	PUNCT
ejpam-6015	79	1	=	=	SYM
ejpam-6015	79	2	0	0	NUM
ejpam-6015	79	3	,	,	PUNCT
ejpam-6015	79	4	i.e	i.e	PRON
ejpam-6015	79	5	θ(sinh(δ(a	θ(sinh(δ(a	NOUN
ejpam-6015	79	6	,	,	PUNCT
ejpam-6015	79	7	ξ	ξ	NOUN
ejpam-6015	79	8	)	)	PUNCT
ejpam-6015	79	9	)	)	PUNCT
ejpam-6015	79	10	)	)	PUNCT
ejpam-6015	80	1	=	=	SYM
ejpam-6015	80	2	0	0	NUM
ejpam-6015	80	3	∀a	∀a	NOUN
ejpam-6015	80	4	∈	∈	PROPN
ejpam-6015	80	5	π	π	X
ejpam-6015	80	6	.	.	PUNCT
ejpam-6015	81	1	then	then	ADV
ejpam-6015	81	2	for	for	ADP
ejpam-6015	81	3	all	all	DET
ejpam-6015	81	4	a	a	DET
ejpam-6015	81	5	∈	∈	PROPN
ejpam-6015	81	6	π	π	NOUN
ejpam-6015	81	7	,	,	PUNCT
ejpam-6015	81	8	∃(bn	∃(bn	PROPN
ejpam-6015	81	9	)	)	PUNCT
ejpam-6015	81	10	⊂	⊂	X
ejpam-6015	82	1	ξ	ξ	PROPN
ejpam-6015	83	1	so	so	ADV
ejpam-6015	83	2	that	that	SCONJ
ejpam-6015	83	3	lim	lim	PROPN
ejpam-6015	83	4	n→+∞	n→+∞	PROPN
ejpam-6015	83	5	θ(sinh(d(a	θ(sinh(d(a	PROPN
ejpam-6015	83	6	,	,	PUNCT
ejpam-6015	83	7	bn	bn	NOUN
ejpam-6015	83	8	)	)	PUNCT
ejpam-6015	83	9	)	)	PUNCT
ejpam-6015	83	10	)	)	PUNCT
ejpam-6015	84	1	=	=	PUNCT
ejpam-6015	84	2	0	0	X
ejpam-6015	84	3	.	.	PUNCT
ejpam-6015	85	1	by	by	ADP
ejpam-6015	85	2	(	(	PUNCT
ejpam-6015	85	3	1	1	NUM
ejpam-6015	85	4	)	)	PUNCT
ejpam-6015	85	5	,	,	PUNCT
ejpam-6015	85	6	we	we	PRON
ejpam-6015	85	7	obtain	obtain	VERB
ejpam-6015	85	8	lim	lim	PROPN
ejpam-6015	85	9	n→+∞	n→+∞	PROPN
ejpam-6015	85	10	(	(	PUNCT
ejpam-6015	85	11	sinh(d(a	sinh(d(a	PROPN
ejpam-6015	85	12	,	,	PUNCT
ejpam-6015	85	13	bn	bn	NOUN
ejpam-6015	85	14	)	)	PUNCT
ejpam-6015	85	15	)	)	PUNCT
ejpam-6015	85	16	)	)	PUNCT
ejpam-6015	85	17	τ	τ	X
ejpam-6015	86	1	=	=	SYM
ejpam-6015	86	2	0	0	NUM
ejpam-6015	86	3	,	,	PUNCT
ejpam-6015	86	4	for	for	ADP
ejpam-6015	86	5	some	some	DET
ejpam-6015	86	6	τ	τ	PROPN
ejpam-6015	86	7	>	>	X
ejpam-6015	86	8	0	0	PROPN
ejpam-6015	86	9	,	,	PUNCT
ejpam-6015	86	10	which	which	PRON
ejpam-6015	86	11	implies	imply	VERB
ejpam-6015	86	12	lim	lim	PROPN
ejpam-6015	86	13	n→+∞	n→+∞	PROPN
ejpam-6015	86	14	sinh(d(a	sinh(d(a	PROPN
ejpam-6015	86	15	,	,	PUNCT
ejpam-6015	86	16	bn	bn	NOUN
ejpam-6015	86	17	)	)	PUNCT
ejpam-6015	86	18	)	)	PUNCT
ejpam-6015	87	1	=	=	PUNCT
ejpam-6015	87	2	0	0	X
ejpam-6015	87	3	.	.	PUNCT
ejpam-6015	88	1	thus	thus	ADV
ejpam-6015	88	2	,	,	PUNCT
ejpam-6015	88	3	for	for	ADP
ejpam-6015	88	4	all	all	DET
ejpam-6015	88	5	a	a	DET
ejpam-6015	88	6	∈	∈	PROPN
ejpam-6015	88	7	π	π	NOUN
ejpam-6015	88	8	,	,	PUNCT
ejpam-6015	88	9	lim	lim	PROPN
ejpam-6015	88	10	n→+∞	n→+∞	VERB
ejpam-6015	88	11	δ(a	δ(a	PROPN
ejpam-6015	88	12	,	,	PUNCT
ejpam-6015	88	13	bn	bn	NOUN
ejpam-6015	88	14	)	)	PUNCT
ejpam-6015	88	15	=	=	SYM
ejpam-6015	88	16	0	0	NUM
ejpam-6015	88	17	,	,	PUNCT
ejpam-6015	88	18	i.e	i.e	PRON
ejpam-6015	88	19	a	a	DET
ejpam-6015	88	20	∈	∈	NOUN
ejpam-6015	88	21	ξ	ξ	X
ejpam-6015	88	22	=	=	SYM
ejpam-6015	88	23	ξ	ξ	PROPN
ejpam-6015	88	24	.	.	PUNCT
ejpam-6015	89	1	so	so	ADV
ejpam-6015	89	2	,	,	PUNCT
ejpam-6015	89	3	π	π	PROPN
ejpam-6015	89	4	⊂	⊂	PROPN
ejpam-6015	89	5	ξ	ξ	PROPN
ejpam-6015	89	6	.	.	PROPN
ejpam-6015	89	7	similarly	similarly	ADV
ejpam-6015	89	8	,	,	PUNCT
ejpam-6015	89	9	as	as	ADP
ejpam-6015	89	10	∆θ(ξ	∆θ(ξ	PROPN
ejpam-6015	89	11	,	,	PUNCT
ejpam-6015	89	12	π	π	X
ejpam-6015	89	13	)	)	PUNCT
ejpam-6015	89	14	=	=	SYM
ejpam-6015	89	15	0	0	NUM
ejpam-6015	89	16	,	,	PUNCT
ejpam-6015	89	17	we	we	PRON
ejpam-6015	89	18	have	have	VERB
ejpam-6015	89	19	ξ	ξ	PROPN
ejpam-6015	89	20	⊂	⊂	PROPN
ejpam-6015	89	21	π	π	PROPN
ejpam-6015	89	22	.	.	PUNCT
ejpam-6015	90	1	finally	finally	ADV
ejpam-6015	90	2	,	,	PUNCT
ejpam-6015	90	3	we	we	PRON
ejpam-6015	90	4	obtain	obtain	VERB
ejpam-6015	90	5	π	π	NOUN
ejpam-6015	90	6	=	=	SYM
ejpam-6015	90	7	ξ	ξ	PROPN
ejpam-6015	90	8	.	.	PUNCT
ejpam-6015	90	9	(	(	PUNCT
ejpam-6015	90	10	ii	ii	NOUN
ejpam-6015	90	11	)	)	PUNCT
ejpam-6015	90	12	if	if	SCONJ
ejpam-6015	90	13	θ(0	θ(0	PROPN
ejpam-6015	90	14	)	)	PUNCT
ejpam-6015	90	15	=	=	SYM
ejpam-6015	90	16	0	0	NUM
ejpam-6015	90	17	,	,	PUNCT
ejpam-6015	90	18	then	then	ADV
ejpam-6015	90	19	hθ(π	hθ(π	NUM
ejpam-6015	90	20	,	,	PUNCT
ejpam-6015	90	21	π	π	X
ejpam-6015	90	22	)	)	PUNCT
ejpam-6015	90	23	=	=	SYM
ejpam-6015	90	24	∆θ(π	∆θ(π	PROPN
ejpam-6015	90	25	,	,	PUNCT
ejpam-6015	90	26	π	π	X
ejpam-6015	90	27	)	)	PUNCT
ejpam-6015	90	28	=	=	SYM
ejpam-6015	90	29	sup	sup	NOUN
ejpam-6015	90	30	a∈π	a∈π	PROPN
ejpam-6015	90	31	inf	inf	PROPN
ejpam-6015	90	32	b∈ξ	b∈ξ	PROPN
ejpam-6015	90	33	θ(sinh(d(a	θ(sinh(d(a	PROPN
ejpam-6015	90	34	,	,	PUNCT
ejpam-6015	90	35	b	b	NOUN
ejpam-6015	90	36	)	)	PUNCT
ejpam-6015	90	37	)	)	PUNCT
ejpam-6015	90	38	)	)	PUNCT
ejpam-6015	90	39	≤	≤	NUM
ejpam-6015	90	40	sup	sup	NOUN
ejpam-6015	90	41	a∈π	a∈π	NOUN
ejpam-6015	90	42	θ(sinh(d(a	θ(sinh(d(a	PROPN
ejpam-6015	90	43	,	,	PUNCT
ejpam-6015	90	44	a	a	PRON
ejpam-6015	90	45	)	)	PUNCT
ejpam-6015	90	46	)	)	PUNCT
ejpam-6015	90	47	)	)	PUNCT
ejpam-6015	91	1	=	=	PUNCT
ejpam-6015	91	2	sup	sup	NOUN
ejpam-6015	91	3	a∈π	a∈π	NOUN
ejpam-6015	91	4	θ(sinh(0	θ(sinh(0	NOUN
ejpam-6015	91	5	)	)	PUNCT
ejpam-6015	91	6	)	)	PUNCT
ejpam-6015	92	1	=	=	PUNCT
ejpam-6015	92	2	sup	sup	NUM
ejpam-6015	92	3	a∈π	a∈π	NOUN
ejpam-6015	92	4	θ(0	θ(0	PROPN
ejpam-6015	92	5	)	)	PUNCT
ejpam-6015	92	6	=	=	SYM
ejpam-6015	92	7	sup	sup	NOUN
ejpam-6015	92	8	a∈π	a∈π	NOUN
ejpam-6015	92	9	(	(	PUNCT
ejpam-6015	92	10	0	0	NUM
ejpam-6015	92	11	)	)	PUNCT
ejpam-6015	92	12	=	=	SYM
ejpam-6015	92	13	0	0	X
ejpam-6015	92	14	.	.	PUNCT
ejpam-6015	93	1	h.	h.	PROPN
ejpam-6015	93	2	aydi	aydi	VERB
ejpam-6015	93	3	et	et	PROPN
ejpam-6015	93	4	al	al	PROPN
ejpam-6015	93	5	.	.	PUNCT
ejpam-6015	93	6	/	/	SYM
ejpam-6015	93	7	eur	eur	PROPN
ejpam-6015	93	8	.	.	PUNCT
ejpam-6015	94	1	j.	j.	PROPN
ejpam-6015	94	2	pure	pure	PROPN
ejpam-6015	94	3	appl	appl	PROPN
ejpam-6015	94	4	.	.	PROPN
ejpam-6015	94	5	math	math	PROPN
ejpam-6015	94	6	,	,	PUNCT
ejpam-6015	94	7	18	18	NUM
ejpam-6015	94	8	(	(	PUNCT
ejpam-6015	94	9	2	2	NUM
ejpam-6015	94	10	)	)	PUNCT
ejpam-6015	94	11	(	(	PUNCT
ejpam-6015	94	12	2025	2025	NUM
ejpam-6015	94	13	)	)	PUNCT
ejpam-6015	94	14	,	,	PUNCT
ejpam-6015	94	15	6015	6015	NUM
ejpam-6015	94	16	5	5	NUM
ejpam-6015	94	17	of	of	ADP
ejpam-6015	94	18	14	14	NUM
ejpam-6015	94	19	thus	thus	ADV
ejpam-6015	94	20	,	,	PUNCT
ejpam-6015	94	21	hθ(π	hθ(π	NUM
ejpam-6015	94	22	,	,	PUNCT
ejpam-6015	94	23	π	π	X
ejpam-6015	94	24	)	)	PUNCT
ejpam-6015	94	25	=	=	SYM
ejpam-6015	95	1	0	0	X
ejpam-6015	95	2	.	.	PUNCT
ejpam-6015	95	3	(	(	PUNCT
ejpam-6015	95	4	iii	iii	X
ejpam-6015	95	5	)	)	PUNCT
ejpam-6015	95	6	it	it	PRON
ejpam-6015	95	7	is	be	AUX
ejpam-6015	95	8	obvious	obvious	ADJ
ejpam-6015	95	9	.	.	PUNCT
ejpam-6015	96	1	notice	notice	VERB
ejpam-6015	96	2	that	that	SCONJ
ejpam-6015	96	3	a	a	DET
ejpam-6015	96	4	hausdorff	hausdorff	NOUN
ejpam-6015	96	5	θ	θ	ADJ
ejpam-6015	96	6	-	-	ADJ
ejpam-6015	96	7	hyperbolic	hyperbolic	ADJ
ejpam-6015	96	8	sine	sine	NOUN
ejpam-6015	96	9	distance	distance	NOUN
ejpam-6015	96	10	function	function	NOUN
ejpam-6015	96	11	is	be	AUX
ejpam-6015	96	12	not	not	PART
ejpam-6015	96	13	necessarily	necessarily	ADV
ejpam-6015	96	14	a	a	DET
ejpam-6015	96	15	hausdorff	hausdorff	NOUN
ejpam-6015	96	16	metric	metric	ADJ
ejpam-6015	96	17	,	,	PUNCT
ejpam-6015	96	18	even	even	ADV
ejpam-6015	96	19	θ(0	θ(0	PROPN
ejpam-6015	96	20	)	)	PUNCT
ejpam-6015	96	21	=	=	SYM
ejpam-6015	97	1	0	0	X
ejpam-6015	97	2	.	.	PUNCT
ejpam-6015	98	1	the	the	DET
ejpam-6015	98	2	following	follow	VERB
ejpam-6015	98	3	example	example	NOUN
ejpam-6015	98	4	shows	show	VERB
ejpam-6015	98	5	this	this	DET
ejpam-6015	98	6	fact	fact	NOUN
ejpam-6015	98	7	.	.	PUNCT
ejpam-6015	99	1	example	example	NOUN
ejpam-6015	100	1	2	2	NUM
ejpam-6015	100	2	.	.	PUNCT
ejpam-6015	100	3	let	let	VERB
ejpam-6015	100	4	x	x	PUNCT
ejpam-6015	100	5	=	=	SYM
ejpam-6015	100	6	r	r	NOUN
ejpam-6015	100	7	and	and	CCONJ
ejpam-6015	100	8	d(ϖ	d(ϖ	PROPN
ejpam-6015	100	9	,	,	PUNCT
ejpam-6015	100	10	ς	ς	NOUN
ejpam-6015	100	11	)	)	PUNCT
ejpam-6015	100	12	=	=	PRON
ejpam-6015	100	13	|ς	|ς	ADJ
ejpam-6015	100	14	−	−	PROPN
ejpam-6015	101	1	ϖ|	ϖ|	ADV
ejpam-6015	101	2	for	for	ADP
ejpam-6015	101	3	all	all	DET
ejpam-6015	101	4	ϖ	ϖ	NOUN
ejpam-6015	101	5	,	,	PUNCT
ejpam-6015	101	6	ς	ς	PROPN
ejpam-6015	101	7	∈	∈	PROPN
ejpam-6015	101	8	x.	x.	NOUN
ejpam-6015	101	9	let	let	VERB
ejpam-6015	101	10	θ(t	θ(t	PROPN
ejpam-6015	101	11	)	)	PUNCT
ejpam-6015	102	1	=	=	SYM
ejpam-6015	102	2	t	t	NOUN
ejpam-6015	102	3	for	for	ADP
ejpam-6015	102	4	any	any	DET
ejpam-6015	102	5	t	t	PROPN
ejpam-6015	102	6	≥	≥	NOUN
ejpam-6015	102	7	0	0	NUM
ejpam-6015	102	8	.	.	PUNCT
ejpam-6015	103	1	take	take	VERB
ejpam-6015	103	2	a	a	DET
ejpam-6015	103	3	=	=	PUNCT
ejpam-6015	103	4	{	{	PUNCT
ejpam-6015	103	5	0	0	NUM
ejpam-6015	103	6	}	}	PUNCT
ejpam-6015	103	7	,	,	PUNCT
ejpam-6015	103	8	b	b	X
ejpam-6015	103	9	=	=	SYM
ejpam-6015	103	10	{	{	PUNCT
ejpam-6015	103	11	2n	2n	NUM
ejpam-6015	103	12	}	}	PUNCT
ejpam-6015	103	13	and	and	CCONJ
ejpam-6015	103	14	c	c	NOUN
ejpam-6015	103	15	=	=	SYM
ejpam-6015	103	16	{	{	PUNCT
ejpam-6015	103	17	n	n	CCONJ
ejpam-6015	103	18	}	}	PUNCT
ejpam-6015	103	19	,	,	PUNCT
ejpam-6015	103	20	with	with	ADP
ejpam-6015	103	21	n	n	PRON
ejpam-6015	103	22	≥	≥	NUM
ejpam-6015	103	23	1	1	NUM
ejpam-6015	103	24	.	.	PUNCT
ejpam-6015	104	1	we	we	PRON
ejpam-6015	104	2	write	write	VERB
ejpam-6015	104	3	hθ(π	hθ(π	NOUN
ejpam-6015	104	4	,	,	PUNCT
ejpam-6015	104	5	ξ	ξ	NOUN
ejpam-6015	104	6	)	)	PUNCT
ejpam-6015	104	7	hθ(π	hθ(π	NOUN
ejpam-6015	104	8	,	,	PUNCT
ejpam-6015	104	9	c	c	X
ejpam-6015	104	10	)	)	PUNCT
ejpam-6015	105	1	+	+	ADV
ejpam-6015	105	2	hθ(c	hθ(c	NUM
ejpam-6015	105	3	,	,	PUNCT
ejpam-6015	105	4	ξ	ξ	NOUN
ejpam-6015	105	5	)	)	PUNCT
ejpam-6015	105	6	=	=	SYM
ejpam-6015	105	7	θ(sinh(2n	θ(sinh(2n	NUM
ejpam-6015	105	8	)	)	PUNCT
ejpam-6015	105	9	)	)	PUNCT
ejpam-6015	106	1	2θ(sinh(n	2θ(sinh(n	NUM
ejpam-6015	106	2	)	)	PUNCT
ejpam-6015	106	3	)	)	PUNCT
ejpam-6015	107	1	=	=	SYM
ejpam-6015	107	2	1	1	NUM
ejpam-6015	107	3	2	2	NUM
ejpam-6015	107	4	(	(	PUNCT
ejpam-6015	107	5	en	en	X
ejpam-6015	107	6	+	+	X
ejpam-6015	107	7	e−n	e−n	NOUN
ejpam-6015	107	8	)	)	PUNCT
ejpam-6015	107	9	→	→	PUNCT
ejpam-6015	108	1	+	+	PROPN
ejpam-6015	108	2	+	+	PROPN
ejpam-6015	108	3	∞	∞	PROPN
ejpam-6015	108	4	asn→	asn→	X
ejpam-6015	108	5	+	+	NOUN
ejpam-6015	108	6	∞	∞	PROPN
ejpam-6015	108	7	,	,	PUNCT
ejpam-6015	108	8	which	which	PRON
ejpam-6015	108	9	shows	show	VERB
ejpam-6015	108	10	that	that	SCONJ
ejpam-6015	108	11	hθ	hθ	PROPN
ejpam-6015	108	12	does	do	AUX
ejpam-6015	108	13	not	not	PART
ejpam-6015	108	14	satisfy	satisfy	VERB
ejpam-6015	108	15	the	the	DET
ejpam-6015	108	16	triangle	triangle	NOUN
ejpam-6015	108	17	inequality	inequality	NOUN
ejpam-6015	108	18	,	,	PUNCT
ejpam-6015	108	19	and	and	CCONJ
ejpam-6015	108	20	so	so	ADV
ejpam-6015	108	21	hθ	hθ	INTJ
ejpam-6015	108	22	is	be	AUX
ejpam-6015	108	23	not	not	PART
ejpam-6015	108	24	a	a	DET
ejpam-6015	108	25	hausdorff	hausdorff	NOUN
ejpam-6015	108	26	metric	metric	ADJ
ejpam-6015	108	27	on	on	ADP
ejpam-6015	108	28	cb(x	cb(x	NUM
ejpam-6015	108	29	)	)	PUNCT
ejpam-6015	108	30	.	.	PUNCT
ejpam-6015	109	1	definition	definition	NOUN
ejpam-6015	109	2	4	4	X
ejpam-6015	109	3	.	.	PUNCT
ejpam-6015	110	1	let	let	VERB
ejpam-6015	110	2	(	(	PUNCT
ejpam-6015	110	3	x	x	NOUN
ejpam-6015	110	4	,	,	PUNCT
ejpam-6015	110	5	d	d	PROPN
ejpam-6015	110	6	)	)	PUNCT
ejpam-6015	110	7	a	a	DET
ejpam-6015	110	8	ms	ms	PROPN
ejpam-6015	110	9	.	.	PROPN
ejpam-6015	111	1	a	a	DET
ejpam-6015	111	2	function	function	NOUN
ejpam-6015	111	3	f	f	NOUN
ejpam-6015	111	4	:	:	PUNCT
ejpam-6015	112	1	x	x	X
ejpam-6015	112	2	→	→	PUNCT
ejpam-6015	112	3	[	[	X
ejpam-6015	112	4	0,+∞	0,+∞	NUM
ejpam-6015	112	5	)	)	PUNCT
ejpam-6015	112	6	is	be	AUX
ejpam-6015	112	7	termed	term	VERB
ejpam-6015	112	8	as	as	ADP
ejpam-6015	112	9	lower	low	ADJ
ejpam-6015	112	10	semicontinuous	semicontinuous	ADJ
ejpam-6015	112	11	if	if	SCONJ
ejpam-6015	112	12	for	for	ADP
ejpam-6015	112	13	{	{	PUNCT
ejpam-6015	112	14	ςn	ςn	NOUN
ejpam-6015	112	15	}	}	PUNCT
ejpam-6015	112	16	⊂	⊂	PROPN
ejpam-6015	112	17	x	x	X
ejpam-6015	112	18	and	and	CCONJ
ejpam-6015	112	19	ς	ς	PROPN
ejpam-6015	112	20	∈	∈	PROPN
ejpam-6015	112	21	x	x	X
ejpam-6015	112	22	,	,	PUNCT
ejpam-6015	112	23	we	we	PRON
ejpam-6015	112	24	have	have	VERB
ejpam-6015	112	25	lim	lim	PROPN
ejpam-6015	112	26	n→+∞	n→+∞	PROPN
ejpam-6015	112	27	d(ςn	d(ςn	PROPN
ejpam-6015	112	28	,	,	PUNCT
ejpam-6015	112	29	ς	ς	NOUN
ejpam-6015	112	30	)	)	PUNCT
ejpam-6015	112	31	=	=	SYM
ejpam-6015	112	32	0	0	NUM
ejpam-6015	113	1	⇒	⇒	NOUN
ejpam-6015	113	2	f(ς	f(ς	PROPN
ejpam-6015	113	3	)	)	PUNCT
ejpam-6015	113	4	≤	≤	PROPN
ejpam-6015	113	5	lim	lim	PROPN
ejpam-6015	113	6	inf	inf	PROPN
ejpam-6015	113	7	n→+∞	n→+∞	PROPN
ejpam-6015	113	8	f(ςn	f(ςn	PROPN
ejpam-6015	113	9	)	)	PUNCT
ejpam-6015	113	10	.	.	PUNCT
ejpam-6015	114	1	for	for	ADP
ejpam-6015	114	2	t	t	PROPN
ejpam-6015	114	3	:	:	PUNCT
ejpam-6015	114	4	x	x	X
ejpam-6015	114	5	→	→	X
ejpam-6015	114	6	cb(x	cb(x	NUM
ejpam-6015	114	7	)	)	PUNCT
ejpam-6015	114	8	,	,	PUNCT
ejpam-6015	114	9	define	define	VERB
ejpam-6015	114	10	ft	ft	X
ejpam-6015	114	11	:	:	PUNCT
ejpam-6015	114	12	x	x	X
ejpam-6015	114	13	→	→	PUNCT
ejpam-6015	115	1	[	[	X
ejpam-6015	115	2	0,+∞	0,+∞	NUM
ejpam-6015	115	3	)	)	PUNCT
ejpam-6015	115	4	by	by	ADP
ejpam-6015	115	5	ft	ft	X
ejpam-6015	115	6	(	(	PUNCT
ejpam-6015	115	7	ς	ς	NOUN
ejpam-6015	115	8	)	)	PUNCT
ejpam-6015	115	9	=	=	SYM
ejpam-6015	115	10	d(ς	d(ς	NOUN
ejpam-6015	115	11	,	,	PUNCT
ejpam-6015	115	12	t	t	PROPN
ejpam-6015	115	13	ς	ς	PROPN
ejpam-6015	115	14	)	)	PUNCT
ejpam-6015	115	15	for	for	ADP
ejpam-6015	115	16	all	all	PRON
ejpam-6015	115	17	ς	ς	PROPN
ejpam-6015	115	18	∈	∈	PROPN
ejpam-6015	115	19	x.	x.	NOUN
ejpam-6015	116	1	3	3	X
ejpam-6015	116	2	.	.	PUNCT
ejpam-6015	116	3	fp	fp	PROPN
ejpam-6015	116	4	results	result	NOUN
ejpam-6015	116	5	in	in	ADP
ejpam-6015	116	6	this	this	DET
ejpam-6015	116	7	part	part	NOUN
ejpam-6015	116	8	,	,	PUNCT
ejpam-6015	116	9	we	we	PRON
ejpam-6015	116	10	present	present	VERB
ejpam-6015	116	11	fp	fp	X
ejpam-6015	116	12	results	result	NOUN
ejpam-6015	116	13	for	for	ADP
ejpam-6015	116	14	some	some	DET
ejpam-6015	116	15	multivalued	multivalued	ADJ
ejpam-6015	116	16	contractions	contraction	NOUN
ejpam-6015	116	17	via	via	ADP
ejpam-6015	116	18	θ	θ	ADJ
ejpam-6015	116	19	-	-	ADJ
ejpam-6015	116	20	hyperbolic	hyperbolic	ADJ
ejpam-6015	116	21	sine	sine	NOUN
ejpam-6015	116	22	functions	function	NOUN
ejpam-6015	116	23	.	.	PUNCT
ejpam-6015	117	1	3.1	3.1	NUM
ejpam-6015	117	2	.	.	NOUN
ejpam-6015	117	3	multivalued	multivalue	VERB
ejpam-6015	117	4	θ	θ	ADJ
ejpam-6015	117	5	-	-	ADJ
ejpam-6015	117	6	hyperbolic	hyperbolic	ADJ
ejpam-6015	117	7	contractions	contraction	NOUN
ejpam-6015	117	8	jleli	jleli	VERB
ejpam-6015	117	9	and	and	CCONJ
ejpam-6015	117	10	samet	samet	VERB
ejpam-6015	118	1	[	[	X
ejpam-6015	118	2	1	1	NUM
ejpam-6015	118	3	]	]	PUNCT
ejpam-6015	118	4	introduced	introduce	VERB
ejpam-6015	118	5	the	the	DET
ejpam-6015	118	6	following	follow	VERB
ejpam-6015	118	7	class	class	NOUN
ejpam-6015	118	8	of	of	ADP
ejpam-6015	118	9	single	single	ADJ
ejpam-6015	118	10	-	-	PUNCT
ejpam-6015	118	11	valued	value	VERB
ejpam-6015	118	12	mappings	mapping	NOUN
ejpam-6015	118	13	.	.	PUNCT
ejpam-6015	119	1	definition	definition	NOUN
ejpam-6015	119	2	5	5	NUM
ejpam-6015	119	3	.	.	PUNCT
ejpam-6015	120	1	let	let	VERB
ejpam-6015	120	2	(	(	PUNCT
ejpam-6015	120	3	x	x	NOUN
ejpam-6015	120	4	,	,	PUNCT
ejpam-6015	120	5	d	d	NOUN
ejpam-6015	120	6	)	)	PUNCT
ejpam-6015	120	7	be	be	AUX
ejpam-6015	120	8	a	a	DET
ejpam-6015	120	9	ms	ms	NOUN
ejpam-6015	120	10	and	and	CCONJ
ejpam-6015	120	11	θ	θ	PROPN
ejpam-6015	120	12	∈	∈	PROPN
ejpam-6015	120	13	θτ	θτ	NOUN
ejpam-6015	120	14	for	for	ADP
ejpam-6015	120	15	some	some	PRON
ejpam-6015	120	16	τ	τ	PROPN
ejpam-6015	120	17	>	>	X
ejpam-6015	120	18	0	0	PROPN
ejpam-6015	120	19	.	.	PUNCT
ejpam-6015	121	1	a	a	DET
ejpam-6015	121	2	mapping	mapping	NOUN
ejpam-6015	121	3	t	t	NOUN
ejpam-6015	121	4	:	:	PUNCT
ejpam-6015	121	5	x	x	X
ejpam-6015	121	6	→	→	PUNCT
ejpam-6015	121	7	x	x	X
ejpam-6015	121	8	is	be	AUX
ejpam-6015	121	9	called	call	VERB
ejpam-6015	121	10	a	a	DET
ejpam-6015	121	11	θ	θ	ADJ
ejpam-6015	121	12	-	-	ADJ
ejpam-6015	121	13	hyperbolic	hyperbolic	ADJ
ejpam-6015	121	14	contraction	contraction	NOUN
ejpam-6015	121	15	on	on	ADP
ejpam-6015	121	16	x	x	NOUN
ejpam-6015	121	17	,	,	PUNCT
ejpam-6015	121	18	if	if	SCONJ
ejpam-6015	121	19	there	there	PRON
ejpam-6015	121	20	is	be	VERB
ejpam-6015	121	21	k	k	PROPN
ejpam-6015	121	22	∈	∈	PROPN
ejpam-6015	121	23	(	(	PUNCT
ejpam-6015	121	24	0	0	NUM
ejpam-6015	121	25	,	,	PUNCT
ejpam-6015	121	26	1	1	NUM
ejpam-6015	121	27	)	)	PUNCT
ejpam-6015	121	28	so	so	SCONJ
ejpam-6015	121	29	that	that	SCONJ
ejpam-6015	121	30	dθ(tς	dθ(tς	PROPN
ejpam-6015	121	31	,	,	PUNCT
ejpam-6015	121	32	tϖ	tϖ	NOUN
ejpam-6015	121	33	)	)	PUNCT
ejpam-6015	121	34	≤	≤	PROPN
ejpam-6015	121	35	kdθ(ϖ	kdθ(ϖ	PROPN
ejpam-6015	121	36	,	,	PUNCT
ejpam-6015	121	37	ς	ς	NOUN
ejpam-6015	121	38	)	)	PUNCT
ejpam-6015	121	39	(	(	PUNCT
ejpam-6015	121	40	2	2	X
ejpam-6015	121	41	)	)	PUNCT
ejpam-6015	121	42	for	for	ADP
ejpam-6015	121	43	all	all	DET
ejpam-6015	121	44	ϖ	ϖ	PROPN
ejpam-6015	121	45	,	,	PUNCT
ejpam-6015	121	46	ς	ς	PROPN
ejpam-6015	121	47	∈	∈	PROPN
ejpam-6015	121	48	x.	x.	NOUN
ejpam-6015	121	49	also	also	ADV
ejpam-6015	121	50	,	,	PUNCT
ejpam-6015	121	51	they	they	PRON
ejpam-6015	121	52	established	establish	VERB
ejpam-6015	121	53	the	the	DET
ejpam-6015	121	54	following	follow	VERB
ejpam-6015	121	55	fp	fp	X
ejpam-6015	121	56	theorem	theorem	PROPN
ejpam-6015	121	57	.	.	PROPN
ejpam-6015	121	58	theorem	theorem	NOUN
ejpam-6015	121	59	2	2	NUM
ejpam-6015	121	60	.	.	PUNCT
ejpam-6015	122	1	[	[	X
ejpam-6015	122	2	1	1	X
ejpam-6015	122	3	]	]	X
ejpam-6015	122	4	let	let	VERB
ejpam-6015	122	5	(	(	PUNCT
ejpam-6015	122	6	x	x	NOUN
ejpam-6015	122	7	,	,	PUNCT
ejpam-6015	122	8	d	d	NOUN
ejpam-6015	122	9	)	)	PUNCT
ejpam-6015	122	10	be	be	AUX
ejpam-6015	122	11	a	a	DET
ejpam-6015	122	12	complete	complete	ADJ
ejpam-6015	122	13	ms	ms	NOUN
ejpam-6015	122	14	and	and	CCONJ
ejpam-6015	122	15	θ	θ	PROPN
ejpam-6015	122	16	∈	∈	PROPN
ejpam-6015	122	17	θτ	θτ	NOUN
ejpam-6015	122	18	for	for	ADP
ejpam-6015	122	19	some	some	PRON
ejpam-6015	122	20	τ	τ	PROPN
ejpam-6015	122	21	>	>	X
ejpam-6015	122	22	0	0	PROPN
ejpam-6015	122	23	.	.	PUNCT
ejpam-6015	123	1	given	give	VERB
ejpam-6015	123	2	t	t	PROPN
ejpam-6015	123	3	:	:	PUNCT
ejpam-6015	123	4	x	x	X
ejpam-6015	123	5	→	→	PUNCT
ejpam-6015	123	6	x	x	X
ejpam-6015	123	7	so	so	SCONJ
ejpam-6015	123	8	that	that	SCONJ
ejpam-6015	123	9	:	:	PUNCT
ejpam-6015	123	10	(	(	PUNCT
ejpam-6015	123	11	i	i	NOUN
ejpam-6015	123	12	)	)	PUNCT
ejpam-6015	123	13	t	t	PROPN
ejpam-6015	123	14	is	be	AUX
ejpam-6015	123	15	a	a	DET
ejpam-6015	123	16	θ	θ	ADJ
ejpam-6015	123	17	-	-	ADJ
ejpam-6015	123	18	hyperbolic	hyperbolic	ADJ
ejpam-6015	123	19	contraction	contraction	NOUN
ejpam-6015	123	20	on	on	ADP
ejpam-6015	123	21	x	x	PROPN
ejpam-6015	123	22	;	;	PUNCT
ejpam-6015	123	23	h.	h.	PROPN
ejpam-6015	123	24	aydi	aydi	VERB
ejpam-6015	123	25	et	et	PROPN
ejpam-6015	123	26	al	al	PROPN
ejpam-6015	123	27	.	.	PUNCT
ejpam-6015	123	28	/	/	SYM
ejpam-6015	123	29	eur	eur	PROPN
ejpam-6015	123	30	.	.	PUNCT
ejpam-6015	124	1	j.	j.	PROPN
ejpam-6015	124	2	pure	pure	PROPN
ejpam-6015	124	3	appl	appl	PROPN
ejpam-6015	124	4	.	.	PROPN
ejpam-6015	124	5	math	math	PROPN
ejpam-6015	124	6	,	,	PUNCT
ejpam-6015	124	7	18	18	NUM
ejpam-6015	124	8	(	(	PUNCT
ejpam-6015	124	9	2	2	NUM
ejpam-6015	124	10	)	)	PUNCT
ejpam-6015	124	11	(	(	PUNCT
ejpam-6015	124	12	2025	2025	NUM
ejpam-6015	124	13	)	)	PUNCT
ejpam-6015	124	14	,	,	PUNCT
ejpam-6015	124	15	6015	6015	NUM
ejpam-6015	124	16	6	6	NUM
ejpam-6015	124	17	of	of	ADP
ejpam-6015	124	18	14	14	NUM
ejpam-6015	124	19	(	(	PUNCT
ejpam-6015	124	20	ii	ii	NOUN
ejpam-6015	124	21	)	)	PUNCT
ejpam-6015	124	22	for	for	ADP
ejpam-6015	124	23	all	all	DET
ejpam-6015	124	24	ϖ	ϖ	PROPN
ejpam-6015	124	25	,	,	PUNCT
ejpam-6015	124	26	ς	ς	PROPN
ejpam-6015	124	27	∈	∈	PROPN
ejpam-6015	124	28	x	x	NOUN
ejpam-6015	124	29	,	,	PUNCT
ejpam-6015	124	30	if	if	SCONJ
ejpam-6015	124	31	lim	lim	PROPN
ejpam-6015	124	32	n→+∞	n→+∞	PROPN
ejpam-6015	124	33	d(tnϖ	d(tnϖ	PROPN
ejpam-6015	124	34	,	,	PUNCT
ejpam-6015	124	35	ς	ς	NOUN
ejpam-6015	124	36	)	)	PUNCT
ejpam-6015	124	37	=	=	SYM
ejpam-6015	124	38	0	0	NUM
ejpam-6015	124	39	,	,	PUNCT
ejpam-6015	124	40	then	then	ADV
ejpam-6015	124	41	there	there	PRON
ejpam-6015	124	42	exists	exist	VERB
ejpam-6015	124	43	a	a	DET
ejpam-6015	124	44	subsequence	subsequence	NOUN
ejpam-6015	124	45	{	{	PUNCT
ejpam-6015	124	46	tnkς	tnkς	NOUN
ejpam-6015	124	47	}	}	PUNCT
ejpam-6015	124	48	of	of	ADP
ejpam-6015	124	49	{	{	PUNCT
ejpam-6015	124	50	tnς	tnς	NOUN
ejpam-6015	124	51	}	}	PUNCT
ejpam-6015	124	52	such	such	ADJ
ejpam-6015	124	53	that	that	SCONJ
ejpam-6015	124	54	lim	lim	PROPN
ejpam-6015	124	55	k→+∞	k→+∞	PROPN
ejpam-6015	124	56	d(t	d(t	PROPN
ejpam-6015	124	57	(	(	PUNCT
ejpam-6015	124	58	tnkς	tnkς	NOUN
ejpam-6015	124	59	)	)	PUNCT
ejpam-6015	124	60	,	,	PUNCT
ejpam-6015	124	61	tϖ	tϖ	X
ejpam-6015	124	62	)	)	PUNCT
ejpam-6015	124	63	=	=	SYM
ejpam-6015	124	64	0	0	X
ejpam-6015	124	65	.	.	PUNCT
ejpam-6015	125	1	then	then	ADV
ejpam-6015	125	2	,	,	PUNCT
ejpam-6015	125	3	t	t	PROPN
ejpam-6015	125	4	possesses	possess	VERB
ejpam-6015	125	5	one	one	NUM
ejpam-6015	125	6	and	and	CCONJ
ejpam-6015	125	7	only	only	ADV
ejpam-6015	125	8	one	one	NUM
ejpam-6015	125	9	fp	fp	NOUN
ejpam-6015	125	10	.	.	PUNCT
ejpam-6015	126	1	moreover	moreover	ADV
ejpam-6015	126	2	,	,	PUNCT
ejpam-6015	126	3	for	for	ADP
ejpam-6015	126	4	all	all	DET
ejpam-6015	126	5	w0	w0	PROPN
ejpam-6015	126	6	∈	∈	PROPN
ejpam-6015	126	7	x	x	SYM
ejpam-6015	126	8	,	,	PUNCT
ejpam-6015	126	9	{	{	PUNCT
ejpam-6015	126	10	tnw0	tnw0	ADV
ejpam-6015	126	11	}	}	PUNCT
ejpam-6015	126	12	is	be	AUX
ejpam-6015	126	13	convergent	convergent	ADJ
ejpam-6015	126	14	to	to	ADP
ejpam-6015	126	15	this	this	DET
ejpam-6015	126	16	unique	unique	ADJ
ejpam-6015	126	17	fp	fp	NOUN
ejpam-6015	126	18	.	.	PUNCT
ejpam-6015	127	1	we	we	PRON
ejpam-6015	127	2	need	need	VERB
ejpam-6015	127	3	the	the	DET
ejpam-6015	127	4	next	next	ADJ
ejpam-6015	127	5	lemma	lemma	PROPN
ejpam-6015	127	6	for	for	ADP
ejpam-6015	127	7	the	the	DET
ejpam-6015	127	8	rest	rest	NOUN
ejpam-6015	127	9	.	.	PUNCT
ejpam-6015	128	1	lemma	lemma	PROPN
ejpam-6015	128	2	1	1	X
ejpam-6015	128	3	.	.	PUNCT
ejpam-6015	129	1	let	let	VERB
ejpam-6015	129	2	π	π	X
ejpam-6015	129	3	,	,	PUNCT
ejpam-6015	129	4	ξ	ξ	PROPN
ejpam-6015	129	5	∈	∈	PROPN
ejpam-6015	129	6	cb(x	cb(x	NUM
ejpam-6015	129	7	)	)	PUNCT
ejpam-6015	129	8	,	,	PUNCT
ejpam-6015	129	9	a	a	DET
ejpam-6015	129	10	∈	∈	PROPN
ejpam-6015	129	11	π	π	NOUN
ejpam-6015	129	12	and	and	CCONJ
ejpam-6015	129	13	θ	θ	PROPN
ejpam-6015	129	14	∈	∈	PROPN
ejpam-6015	129	15	θτ	θτ	NOUN
ejpam-6015	129	16	for	for	ADP
ejpam-6015	129	17	some	some	PRON
ejpam-6015	129	18	τ	τ	PROPN
ejpam-6015	129	19	>	>	X
ejpam-6015	129	20	0	0	PROPN
ejpam-6015	129	21	.	.	PUNCT
ejpam-6015	130	1	thus	thus	ADV
ejpam-6015	130	2	,	,	PUNCT
ejpam-6015	130	3	for	for	ADP
ejpam-6015	130	4	each	each	DET
ejpam-6015	130	5	ε	ε	PROPN
ejpam-6015	130	6	>	>	X
ejpam-6015	130	7	0	0	PROPN
ejpam-6015	130	8	,	,	PUNCT
ejpam-6015	130	9	there	there	PRON
ejpam-6015	130	10	is	be	VERB
ejpam-6015	130	11	b	b	PROPN
ejpam-6015	130	12	∈	∈	ADP
ejpam-6015	130	13	ξ	ξ	NOUN
ejpam-6015	131	1	so	so	ADV
ejpam-6015	131	2	that	that	DET
ejpam-6015	131	3	dθ(a	dθ(a	NOUN
ejpam-6015	131	4	,	,	PUNCT
ejpam-6015	131	5	b	b	X
ejpam-6015	131	6	)	)	PUNCT
ejpam-6015	131	7	≤	≤	NOUN
ejpam-6015	131	8	hθ(π	hθ(π	NOUN
ejpam-6015	131	9	,	,	PUNCT
ejpam-6015	131	10	ξ	ξ	X
ejpam-6015	131	11	)	)	PUNCT
ejpam-6015	131	12	+	+	CCONJ
ejpam-6015	131	13	ε	ε	PROPN
ejpam-6015	131	14	.	.	PUNCT
ejpam-6015	131	15	proof	proof	NOUN
ejpam-6015	131	16	.	.	PUNCT
ejpam-6015	132	1	let	let	VERB
ejpam-6015	132	2	π	π	X
ejpam-6015	132	3	,	,	PUNCT
ejpam-6015	132	4	ξ	ξ	PROPN
ejpam-6015	132	5	∈	∈	PROPN
ejpam-6015	132	6	cb(x	cb(x	NUM
ejpam-6015	132	7	)	)	PUNCT
ejpam-6015	132	8	and	and	CCONJ
ejpam-6015	132	9	a	a	DET
ejpam-6015	132	10	∈	∈	PROPN
ejpam-6015	132	11	π	π	X
ejpam-6015	132	12	.	.	PUNCT
ejpam-6015	133	1	we	we	PRON
ejpam-6015	133	2	have	have	VERB
ejpam-6015	133	3	dθ(a	dθ(a	PUNCT
ejpam-6015	133	4	,	,	PUNCT
ejpam-6015	133	5	ξ	ξ	NOUN
ejpam-6015	133	6	)	)	PUNCT
ejpam-6015	133	7	≤	≤	NOUN
ejpam-6015	133	8	∆θ(π	∆θ(π	PROPN
ejpam-6015	133	9	,	,	PUNCT
ejpam-6015	133	10	ξ	ξ	NOUN
ejpam-6015	133	11	)	)	PUNCT
ejpam-6015	133	12	≤	≤	NOUN
ejpam-6015	133	13	hθ(π	hθ(π	NOUN
ejpam-6015	133	14	,	,	PUNCT
ejpam-6015	133	15	ξ	ξ	NOUN
ejpam-6015	133	16	)	)	PUNCT
ejpam-6015	133	17	.	.	PUNCT
ejpam-6015	134	1	then	then	ADV
ejpam-6015	134	2	,	,	PUNCT
ejpam-6015	134	3	for	for	ADP
ejpam-6015	134	4	every	every	DET
ejpam-6015	134	5	ε	ε	PROPN
ejpam-6015	134	6	>	>	X
ejpam-6015	134	7	0	0	PROPN
ejpam-6015	134	8	,	,	PUNCT
ejpam-6015	134	9	there	there	PRON
ejpam-6015	134	10	is	be	VERB
ejpam-6015	134	11	b	b	PROPN
ejpam-6015	134	12	∈	∈	ADJ
ejpam-6015	134	13	b	b	NOUN
ejpam-6015	134	14	so	so	ADV
ejpam-6015	134	15	that	that	DET
ejpam-6015	134	16	dθ(a	dθ(a	NOUN
ejpam-6015	134	17	,	,	PUNCT
ejpam-6015	134	18	b	b	X
ejpam-6015	134	19	)	)	PUNCT
ejpam-6015	134	20	≤	≤	NOUN
ejpam-6015	134	21	dθ(a	dθ(a	PUNCT
ejpam-6015	134	22	,	,	PUNCT
ejpam-6015	134	23	ξ	ξ	NOUN
ejpam-6015	134	24	)	)	PUNCT
ejpam-6015	135	1	+	+	CCONJ
ejpam-6015	135	2	ε	ε	PROPN
ejpam-6015	135	3	.	.	PUNCT
ejpam-6015	135	4	consequently	consequently	ADV
ejpam-6015	135	5	,	,	PUNCT
ejpam-6015	135	6	dθ(a	dθ(a	NOUN
ejpam-6015	135	7	,	,	PUNCT
ejpam-6015	135	8	b	b	X
ejpam-6015	135	9	)	)	PUNCT
ejpam-6015	135	10	≤	≤	NOUN
ejpam-6015	135	11	hθ(π	hθ(π	NOUN
ejpam-6015	135	12	,	,	PUNCT
ejpam-6015	135	13	ξ	ξ	X
ejpam-6015	135	14	)	)	PUNCT
ejpam-6015	135	15	+	+	CCONJ
ejpam-6015	136	1	ε	ε	PROPN
ejpam-6015	136	2	.	.	PUNCT
ejpam-6015	137	1	the	the	DET
ejpam-6015	137	2	next	next	ADJ
ejpam-6015	137	3	result	result	NOUN
ejpam-6015	137	4	corresponds	correspond	VERB
ejpam-6015	137	5	to	to	ADP
ejpam-6015	137	6	the	the	DET
ejpam-6015	137	7	extension	extension	NOUN
ejpam-6015	137	8	of	of	ADP
ejpam-6015	137	9	theorem	theorem	NOUN
ejpam-6015	137	10	2	2	NUM
ejpam-6015	137	11	to	to	ADP
ejpam-6015	137	12	multivalued	multivalue	VERB
ejpam-6015	137	13	mappings	mapping	NOUN
ejpam-6015	137	14	.	.	PUNCT
ejpam-6015	138	1	it	it	PRON
ejpam-6015	138	2	is	be	AUX
ejpam-6015	138	3	stated	state	VERB
ejpam-6015	138	4	as	as	SCONJ
ejpam-6015	138	5	follows	follow	VERB
ejpam-6015	138	6	:	:	PUNCT
ejpam-6015	138	7	theorem	theorem	NOUN
ejpam-6015	138	8	3	3	X
ejpam-6015	138	9	.	.	PUNCT
ejpam-6015	139	1	let	let	AUX
ejpam-6015	139	2	(	(	PUNCT
ejpam-6015	139	3	x	x	NOUN
ejpam-6015	139	4	,	,	PUNCT
ejpam-6015	139	5	d	d	NOUN
ejpam-6015	139	6	)	)	PUNCT
ejpam-6015	139	7	be	be	AUX
ejpam-6015	139	8	a	a	DET
ejpam-6015	139	9	complete	complete	ADJ
ejpam-6015	139	10	ms	ms	NOUN
ejpam-6015	139	11	and	and	CCONJ
ejpam-6015	139	12	θ	θ	PROPN
ejpam-6015	139	13	∈	∈	PROPN
ejpam-6015	139	14	θτ	θτ	NOUN
ejpam-6015	139	15	for	for	ADP
ejpam-6015	139	16	τ	τ	PROPN
ejpam-6015	139	17	>	>	X
ejpam-6015	139	18	0	0	X
ejpam-6015	139	19	.	.	PUNCT
ejpam-6015	140	1	let	let	VERB
ejpam-6015	140	2	t	t	NOUN
ejpam-6015	140	3	:	:	PUNCT
ejpam-6015	140	4	x	x	SYM
ejpam-6015	140	5	→	→	X
ejpam-6015	140	6	cb(x	cb(x	NUM
ejpam-6015	140	7	)	)	PUNCT
ejpam-6015	140	8	be	be	AUX
ejpam-6015	140	9	a	a	DET
ejpam-6015	140	10	mapping	mapping	NOUN
ejpam-6015	140	11	such	such	ADJ
ejpam-6015	140	12	that	that	DET
ejpam-6015	140	13	hθ(tς	hθ(tς	PROPN
ejpam-6015	140	14	,	,	PUNCT
ejpam-6015	140	15	tϖ	tϖ	NOUN
ejpam-6015	140	16	)	)	PUNCT
ejpam-6015	140	17	≤	≤	PROPN
ejpam-6015	140	18	kdθ(ϖ	kdθ(ϖ	PROPN
ejpam-6015	140	19	,	,	PUNCT
ejpam-6015	140	20	ς	ς	PROPN
ejpam-6015	140	21	)	)	PUNCT
ejpam-6015	140	22	,	,	PUNCT
ejpam-6015	140	23	(	(	PUNCT
ejpam-6015	140	24	3	3	X
ejpam-6015	140	25	)	)	PUNCT
ejpam-6015	140	26	for	for	ADP
ejpam-6015	140	27	all	all	DET
ejpam-6015	140	28	ϖ	ϖ	PROPN
ejpam-6015	140	29	,	,	PUNCT
ejpam-6015	140	30	ς	ς	PROPN
ejpam-6015	140	31	∈	∈	PROPN
ejpam-6015	141	1	x	x	NOUN
ejpam-6015	141	2	,	,	PUNCT
ejpam-6015	141	3	where	where	SCONJ
ejpam-6015	141	4	k	k	PROPN
ejpam-6015	141	5	∈	∈	PROPN
ejpam-6015	142	1	[	[	X
ejpam-6015	142	2	0	0	NUM
ejpam-6015	142	3	,	,	PUNCT
ejpam-6015	142	4	1	1	NUM
ejpam-6015	142	5	)	)	PUNCT
ejpam-6015	142	6	.	.	PUNCT
ejpam-6015	143	1	assume	assume	VERB
ejpam-6015	143	2	that	that	SCONJ
ejpam-6015	143	3	,	,	PUNCT
ejpam-6015	143	4	ft	ft	NOUN
ejpam-6015	143	5	is	be	AUX
ejpam-6015	143	6	lower	low	ADJ
ejpam-6015	143	7	semi	semi	ADJ
ejpam-6015	143	8	-	-	ADJ
ejpam-6015	143	9	continuous	continuous	ADJ
ejpam-6015	143	10	,	,	PUNCT
ejpam-6015	143	11	then	then	ADV
ejpam-6015	143	12	t	t	PROPN
ejpam-6015	143	13	has	have	VERB
ejpam-6015	143	14	a	a	DET
ejpam-6015	143	15	fp	fp	NOUN
ejpam-6015	143	16	in	in	ADP
ejpam-6015	143	17	x.	x.	NOUN
ejpam-6015	143	18	proof	proof	NOUN
ejpam-6015	143	19	.	.	PUNCT
ejpam-6015	144	1	let	let	VERB
ejpam-6015	144	2	ς0	ς0	PROPN
ejpam-6015	144	3	∈	∈	NOUN
ejpam-6015	144	4	x	x	X
ejpam-6015	144	5	and	and	CCONJ
ejpam-6015	144	6	ς1	ς1	PROPN
ejpam-6015	144	7	∈	∈	PROPN
ejpam-6015	144	8	tς0	tς0	PROPN
ejpam-6015	144	9	.	.	PUNCT
ejpam-6015	145	1	when	when	SCONJ
ejpam-6015	145	2	dθ(ς0	dθ(ς0	NOUN
ejpam-6015	145	3	,	,	PUNCT
ejpam-6015	145	4	ς1	ς1	NOUN
ejpam-6015	145	5	)	)	PUNCT
ejpam-6015	145	6	=	=	SYM
ejpam-6015	145	7	0	0	NUM
ejpam-6015	145	8	,	,	PUNCT
ejpam-6015	145	9	so	so	ADV
ejpam-6015	145	10	by	by	ADP
ejpam-6015	145	11	proposition	proposition	NOUN
ejpam-6015	145	12	1	1	NUM
ejpam-6015	145	13	(	(	PUNCT
ejpam-6015	145	14	i	i	NOUN
ejpam-6015	145	15	)	)	PUNCT
ejpam-6015	145	16	,	,	PUNCT
ejpam-6015	145	17	one	one	PRON
ejpam-6015	145	18	gets	get	VERB
ejpam-6015	145	19	ς0	ς0	NOUN
ejpam-6015	145	20	=	=	PUNCT
ejpam-6015	145	21	ς1	ς1	NOUN
ejpam-6015	145	22	and	and	CCONJ
ejpam-6015	145	23	so	so	ADV
ejpam-6015	145	24	ς0	ς0	PROPN
ejpam-6015	145	25	is	be	AUX
ejpam-6015	145	26	a	a	DET
ejpam-6015	145	27	fp	fp	X
ejpam-6015	145	28	of	of	ADP
ejpam-6015	145	29	t.	t.	PROPN
ejpam-6015	145	30	suppose	suppose	VERB
ejpam-6015	146	1	that	that	SCONJ
ejpam-6015	146	2	dθ(ς0	dθ(ς0	NOUN
ejpam-6015	146	3	,	,	PUNCT
ejpam-6015	146	4	ς1	ς1	NOUN
ejpam-6015	146	5	)	)	PUNCT
ejpam-6015	146	6	>	>	X
ejpam-6015	146	7	0	0	X
ejpam-6015	146	8	.	.	PUNCT
ejpam-6015	147	1	since	since	SCONJ
ejpam-6015	147	2	tς0	tς0	PROPN
ejpam-6015	147	3	,	,	PUNCT
ejpam-6015	147	4	t	t	PROPN
ejpam-6015	147	5	ς1	ς1	PROPN
ejpam-6015	147	6	∈	∈	NOUN
ejpam-6015	147	7	cb(x	cb(x	ADJ
ejpam-6015	147	8	)	)	PUNCT
ejpam-6015	147	9	and	and	CCONJ
ejpam-6015	147	10	ς1	ς1	PROPN
ejpam-6015	147	11	∈	∈	PROPN
ejpam-6015	147	12	tς0	tς0	NOUN
ejpam-6015	147	13	,	,	PUNCT
ejpam-6015	147	14	using	use	VERB
ejpam-6015	147	15	lemma	lemma	PROPN
ejpam-6015	147	16	1	1	NUM
ejpam-6015	147	17	,	,	PUNCT
ejpam-6015	147	18	there	there	PRON
ejpam-6015	147	19	is	be	VERB
ejpam-6015	147	20	ς2	ς2	PROPN
ejpam-6015	147	21	∈	∈	PROPN
ejpam-6015	147	22	tς1	tς1	PROPN
ejpam-6015	147	23	,	,	PUNCT
ejpam-6015	147	24	so	so	SCONJ
ejpam-6015	147	25	that	that	SCONJ
ejpam-6015	147	26	dθ(ς1	dθ(ς1	ADV
ejpam-6015	147	27	,	,	PUNCT
ejpam-6015	147	28	ς2	ς2	PROPN
ejpam-6015	147	29	)	)	PUNCT
ejpam-6015	147	30	≤	≤	NOUN
ejpam-6015	147	31	hθ(tς0	hθ(tς0	PROPN
ejpam-6015	147	32	,	,	PUNCT
ejpam-6015	147	33	t	t	NOUN
ejpam-6015	147	34	ς1	ς1	NOUN
ejpam-6015	147	35	)	)	PUNCT
ejpam-6015	148	1	+	+	CCONJ
ejpam-6015	148	2	1−	1−	NUM
ejpam-6015	148	3	k	k	SYM
ejpam-6015	148	4	2	2	NUM
ejpam-6015	148	5	dθ(ς0	dθ(ς0	NOUN
ejpam-6015	148	6	,	,	PUNCT
ejpam-6015	148	7	ς1	ς1	NOUN
ejpam-6015	148	8	)	)	PUNCT
ejpam-6015	148	9	.	.	PUNCT
ejpam-6015	149	1	(	(	PUNCT
ejpam-6015	149	2	4	4	X
ejpam-6015	149	3	)	)	PUNCT
ejpam-6015	149	4	if	if	SCONJ
ejpam-6015	149	5	dθ(ς1	dθ(ς1	ADV
ejpam-6015	149	6	,	,	PUNCT
ejpam-6015	149	7	ς2	ς2	PROPN
ejpam-6015	149	8	)	)	PUNCT
ejpam-6015	150	1	=	=	SYM
ejpam-6015	150	2	0	0	NUM
ejpam-6015	150	3	,	,	PUNCT
ejpam-6015	150	4	then	then	ADV
ejpam-6015	150	5	by	by	ADP
ejpam-6015	150	6	proposition	proposition	NOUN
ejpam-6015	150	7	1	1	NUM
ejpam-6015	150	8	(	(	PUNCT
ejpam-6015	150	9	i	i	NOUN
ejpam-6015	150	10	)	)	PUNCT
ejpam-6015	150	11	,	,	PUNCT
ejpam-6015	150	12	we	we	PRON
ejpam-6015	150	13	have	have	VERB
ejpam-6015	150	14	ς1	ς1	NOUN
ejpam-6015	150	15	=	=	SYM
ejpam-6015	150	16	ς2	ς2	PROPN
ejpam-6015	150	17	and	and	CCONJ
ejpam-6015	150	18	so	so	ADV
ejpam-6015	150	19	ς1	ς1	NOUN
ejpam-6015	150	20	is	be	AUX
ejpam-6015	150	21	a	a	DET
ejpam-6015	150	22	fp	fp	NOUN
ejpam-6015	150	23	of	of	ADP
ejpam-6015	150	24	t.	t.	PROPN
ejpam-6015	150	25	when	when	SCONJ
ejpam-6015	150	26	dθ(ς1	dθ(ς1	ADV
ejpam-6015	150	27	,	,	PUNCT
ejpam-6015	150	28	ς2	ς2	PROPN
ejpam-6015	150	29	)	)	PUNCT
ejpam-6015	150	30	>	>	X
ejpam-6015	150	31	0	0	NUM
ejpam-6015	150	32	,	,	PUNCT
ejpam-6015	150	33	then	then	ADV
ejpam-6015	150	34	by	by	ADP
ejpam-6015	150	35	lemma	lemma	PROPN
ejpam-6015	150	36	1	1	NUM
ejpam-6015	150	37	,	,	PUNCT
ejpam-6015	150	38	there	there	PRON
ejpam-6015	150	39	is	be	VERB
ejpam-6015	150	40	ς3	ς3	NOUN
ejpam-6015	150	41	∈	∈	PROPN
ejpam-6015	150	42	tς2	tς2	NOUN
ejpam-6015	150	43	,	,	PUNCT
ejpam-6015	150	44	so	so	SCONJ
ejpam-6015	150	45	that	that	SCONJ
ejpam-6015	150	46	dθ(ς2	dθ(ς2	NOUN
ejpam-6015	150	47	,	,	PUNCT
ejpam-6015	150	48	ς3	ς3	NOUN
ejpam-6015	150	49	)	)	PUNCT
ejpam-6015	150	50	≤	≤	NOUN
ejpam-6015	150	51	hθ(tς1	hθ(tς1	PROPN
ejpam-6015	150	52	,	,	PUNCT
ejpam-6015	150	53	t	t	PROPN
ejpam-6015	150	54	ς2	ς2	PROPN
ejpam-6015	150	55	)	)	PUNCT
ejpam-6015	151	1	+	+	NUM
ejpam-6015	151	2	1−	1−	NUM
ejpam-6015	151	3	k	k	NOUN
ejpam-6015	151	4	2	2	NUM
ejpam-6015	151	5	dθ(ς1	dθ(ς1	ADV
ejpam-6015	151	6	,	,	PUNCT
ejpam-6015	151	7	ς2	ς2	PROPN
ejpam-6015	151	8	)	)	PUNCT
ejpam-6015	151	9	.	.	PUNCT
ejpam-6015	152	1	(	(	PUNCT
ejpam-6015	152	2	5	5	X
ejpam-6015	152	3	)	)	PUNCT
ejpam-6015	152	4	continuing	continue	VERB
ejpam-6015	152	5	as	as	ADP
ejpam-6015	152	6	above	above	ADV
ejpam-6015	152	7	,	,	PUNCT
ejpam-6015	152	8	we	we	PRON
ejpam-6015	152	9	construct	construct	VERB
ejpam-6015	152	10	{	{	PUNCT
ejpam-6015	152	11	ςn	ςn	NOUN
ejpam-6015	152	12	}	}	PUNCT
ejpam-6015	152	13	⊂	⊂	NOUN
ejpam-6015	152	14	x	x	PUNCT
ejpam-6015	153	1	so	so	ADV
ejpam-6015	153	2	that	that	SCONJ
ejpam-6015	153	3	ςn+1	ςn+1	NUM
ejpam-6015	153	4	∈	∈	PROPN
ejpam-6015	153	5	t	t	PROPN
ejpam-6015	153	6	(	(	PUNCT
ejpam-6015	153	7	ςn	ςn	NOUN
ejpam-6015	153	8	)	)	PUNCT
ejpam-6015	153	9	with	with	ADP
ejpam-6015	153	10	dθ(ςn	dθ(ςn	PROPN
ejpam-6015	153	11	,	,	PUNCT
ejpam-6015	153	12	ςn+1	ςn+1	NUM
ejpam-6015	153	13	)	)	PUNCT
ejpam-6015	153	14	>	>	SYM
ejpam-6015	153	15	0	0	PUNCT
ejpam-6015	153	16	and	and	CCONJ
ejpam-6015	153	17	dθ(ςn+1	dθ(ςn+1	PROPN
ejpam-6015	153	18	,	,	PUNCT
ejpam-6015	153	19	ςn	ςn	NOUN
ejpam-6015	153	20	)	)	PUNCT
ejpam-6015	153	21	≤	≤	NOUN
ejpam-6015	153	22	hθ(tςn	hθ(tςn	NOUN
ejpam-6015	153	23	,	,	PUNCT
ejpam-6015	153	24	t	t	PROPN
ejpam-6015	153	25	ςn−1	ςn−1	PROPN
ejpam-6015	153	26	)	)	PUNCT
ejpam-6015	153	27	+	+	CCONJ
ejpam-6015	153	28	1−	1−	NUM
ejpam-6015	153	29	k	k	SYM
ejpam-6015	153	30	2	2	NUM
ejpam-6015	153	31	dθ(ςn	dθ(ςn	NOUN
ejpam-6015	153	32	,	,	PUNCT
ejpam-6015	153	33	ςn−1	ςn−1	PROPN
ejpam-6015	153	34	)	)	PUNCT
ejpam-6015	153	35	.	.	PUNCT
ejpam-6015	154	1	h.	h.	PROPN
ejpam-6015	154	2	aydi	aydi	VERB
ejpam-6015	154	3	et	et	PROPN
ejpam-6015	154	4	al	al	PROPN
ejpam-6015	154	5	.	.	PUNCT
ejpam-6015	154	6	/	/	SYM
ejpam-6015	154	7	eur	eur	PROPN
ejpam-6015	154	8	.	.	PUNCT
ejpam-6015	155	1	j.	j.	PROPN
ejpam-6015	155	2	pure	pure	PROPN
ejpam-6015	155	3	appl	appl	PROPN
ejpam-6015	155	4	.	.	PROPN
ejpam-6015	155	5	math	math	PROPN
ejpam-6015	155	6	,	,	PUNCT
ejpam-6015	155	7	18	18	NUM
ejpam-6015	155	8	(	(	PUNCT
ejpam-6015	155	9	2	2	NUM
ejpam-6015	155	10	)	)	PUNCT
ejpam-6015	155	11	(	(	PUNCT
ejpam-6015	155	12	2025	2025	NUM
ejpam-6015	155	13	)	)	PUNCT
ejpam-6015	155	14	,	,	PUNCT
ejpam-6015	155	15	6015	6015	NUM
ejpam-6015	155	16	7	7	NUM
ejpam-6015	155	17	of	of	ADP
ejpam-6015	155	18	14	14	NUM
ejpam-6015	155	19	then	then	ADV
ejpam-6015	155	20	,	,	PUNCT
ejpam-6015	155	21	using	use	VERB
ejpam-6015	155	22	(	(	PUNCT
ejpam-6015	155	23	3	3	NUM
ejpam-6015	155	24	)	)	PUNCT
ejpam-6015	155	25	,	,	PUNCT
ejpam-6015	155	26	we	we	PRON
ejpam-6015	155	27	get	get	VERB
ejpam-6015	155	28	dθ(ςn+1	dθ(ςn+1	ADJ
ejpam-6015	155	29	,	,	PUNCT
ejpam-6015	155	30	ςn	ςn	NOUN
ejpam-6015	155	31	)	)	PUNCT
ejpam-6015	155	32	≤	≤	NOUN
ejpam-6015	155	33	kdθ(ςn	kdθ(ςn	ADJ
ejpam-6015	155	34	,	,	PUNCT
ejpam-6015	155	35	ςn−1	ςn−1	PROPN
ejpam-6015	155	36	)	)	PUNCT
ejpam-6015	156	1	+	+	CCONJ
ejpam-6015	156	2	1−	1−	NUM
ejpam-6015	156	3	k	k	SYM
ejpam-6015	156	4	2	2	NUM
ejpam-6015	156	5	dθ(ςn	dθ(ςn	NOUN
ejpam-6015	156	6	,	,	PUNCT
ejpam-6015	156	7	ςn−1	ςn−1	NOUN
ejpam-6015	156	8	)	)	PUNCT
ejpam-6015	156	9	=	=	SYM
ejpam-6015	157	1	1	1	NUM
ejpam-6015	157	2	+	+	CCONJ
ejpam-6015	157	3	k	k	PROPN
ejpam-6015	157	4	2	2	NUM
ejpam-6015	157	5	dθ(ςn	dθ(ςn	NOUN
ejpam-6015	157	6	,	,	PUNCT
ejpam-6015	157	7	ςn−1	ςn−1	NOUN
ejpam-6015	157	8	)	)	PUNCT
ejpam-6015	157	9	.	.	PUNCT
ejpam-6015	158	1	by	by	ADP
ejpam-6015	158	2	induction	induction	NOUN
ejpam-6015	158	3	,	,	PUNCT
ejpam-6015	158	4	dθ(ςn	dθ(ςn	NOUN
ejpam-6015	158	5	,	,	PUNCT
ejpam-6015	158	6	ςn+1	ςn+1	NUM
ejpam-6015	158	7	)	)	PUNCT
ejpam-6015	158	8	≤	≤	NOUN
ejpam-6015	158	9	(	(	PUNCT
ejpam-6015	158	10	1	1	NUM
ejpam-6015	158	11	+	+	CCONJ
ejpam-6015	158	12	k	k	PROPN
ejpam-6015	158	13	2	2	NUM
ejpam-6015	158	14	)	)	PUNCT
ejpam-6015	158	15	ndθ(ς0	ndθ(ς0	ADV
ejpam-6015	158	16	,	,	PUNCT
ejpam-6015	158	17	ς1	ς1	NOUN
ejpam-6015	158	18	)	)	PUNCT
ejpam-6015	158	19	,	,	PUNCT
ejpam-6015	158	20	∀n	∀n	NUM
ejpam-6015	158	21	≥	≥	NOUN
ejpam-6015	158	22	1	1	NUM
ejpam-6015	158	23	.	.	PUNCT
ejpam-6015	159	1	on	on	ADP
ejpam-6015	159	2	the	the	DET
ejpam-6015	159	3	other	other	ADJ
ejpam-6015	159	4	hand	hand	NOUN
ejpam-6015	159	5	,	,	PUNCT
ejpam-6015	159	6	by	by	ADP
ejpam-6015	159	7	(	(	PUNCT
ejpam-6015	159	8	1	1	NUM
ejpam-6015	159	9	)	)	PUNCT
ejpam-6015	159	10	,	,	PUNCT
ejpam-6015	159	11	and	and	CCONJ
ejpam-6015	159	12	since	since	SCONJ
ejpam-6015	159	13	sinh	sinh	PROPN
ejpam-6015	159	14	t	t	PROPN
ejpam-6015	159	15	≥	≥	PROPN
ejpam-6015	159	16	t	t	PROPN
ejpam-6015	159	17	∀t	∀t	PROPN
ejpam-6015	159	18	≥	≥	NOUN
ejpam-6015	159	19	0	0	NUM
ejpam-6015	159	20	,	,	PUNCT
ejpam-6015	159	21	we	we	PRON
ejpam-6015	159	22	get	get	VERB
ejpam-6015	159	23	θ(sinh(d(ςn	θ(sinh(d(ςn	NOUN
ejpam-6015	159	24	,	,	PUNCT
ejpam-6015	159	25	ςn+1	ςn+1	NUM
ejpam-6015	159	26	)	)	PUNCT
ejpam-6015	159	27	)	)	PUNCT
ejpam-6015	159	28	)	)	PUNCT
ejpam-6015	159	29	≥	≥	NOUN
ejpam-6015	159	30	c(sinh(d(ςn	c(sinh(d(ςn	NUM
ejpam-6015	159	31	,	,	PUNCT
ejpam-6015	159	32	ςn+1	ςn+1	NUM
ejpam-6015	159	33	)	)	PUNCT
ejpam-6015	159	34	)	)	PUNCT
ejpam-6015	159	35	)	)	PUNCT
ejpam-6015	160	1	τ	τ	PROPN
ejpam-6015	160	2	≥	≥	NOUN
ejpam-6015	160	3	c(d(ςn	c(d(ςn	NOUN
ejpam-6015	160	4	,	,	PUNCT
ejpam-6015	160	5	ςn+1	ςn+1	NUM
ejpam-6015	160	6	)	)	PUNCT
ejpam-6015	160	7	)	)	PUNCT
ejpam-6015	161	1	τ	τ	PROPN
ejpam-6015	161	2	,	,	PUNCT
ejpam-6015	161	3	∀n	∀n	NUM
ejpam-6015	161	4	≥	≥	NOUN
ejpam-6015	161	5	1	1	NUM
ejpam-6015	161	6	.	.	PUNCT
ejpam-6015	162	1	this	this	DET
ejpam-6015	162	2	yields	yield	NOUN
ejpam-6015	162	3	to	to	PART
ejpam-6015	162	4	d(ςn	d(ςn	VERB
ejpam-6015	162	5	,	,	PUNCT
ejpam-6015	162	6	ςn+1	ςn+1	NUM
ejpam-6015	162	7	)	)	PUNCT
ejpam-6015	162	8	≤	≤	NOUN
ejpam-6015	162	9	(	(	PUNCT
ejpam-6015	162	10	(	(	PUNCT
ejpam-6015	162	11	1	1	NUM
ejpam-6015	162	12	+	+	CCONJ
ejpam-6015	162	13	k	k	PROPN
ejpam-6015	162	14	2	2	X
ejpam-6015	162	15	)	)	PUNCT
ejpam-6015	162	16	1	1	NUM
ejpam-6015	162	17	τ	τ	X
ejpam-6015	162	18	)	)	PUNCT
ejpam-6015	162	19	n	n	CCONJ
ejpam-6015	162	20	(	(	PUNCT
ejpam-6015	162	21	dθ(ς0	dθ(ς0	PROPN
ejpam-6015	162	22	,	,	PUNCT
ejpam-6015	162	23	ς1	ς1	NOUN
ejpam-6015	162	24	)	)	PUNCT
ejpam-6015	162	25	c	c	NOUN
ejpam-6015	162	26	)	)	PUNCT
ejpam-6015	162	27	1	1	NUM
ejpam-6015	162	28	τ	τ	PROPN
ejpam-6015	162	29	,	,	PUNCT
ejpam-6015	162	30	∀n	∀n	NUM
ejpam-6015	162	31	≥	≥	NOUN
ejpam-6015	162	32	1	1	NUM
ejpam-6015	162	33	.	.	PUNCT
ejpam-6015	163	1	since	since	SCONJ
ejpam-6015	163	2	k	k	PROPN
ejpam-6015	163	3	∈	∈	PROPN
ejpam-6015	163	4	[	[	X
ejpam-6015	163	5	0	0	NUM
ejpam-6015	163	6	,	,	PUNCT
ejpam-6015	163	7	1	1	NUM
ejpam-6015	163	8	)	)	PUNCT
ejpam-6015	163	9	and	and	CCONJ
ejpam-6015	163	10	τ	τ	X
ejpam-6015	163	11	>	>	X
ejpam-6015	163	12	0	0	PROPN
ejpam-6015	163	13	,	,	PUNCT
ejpam-6015	163	14	one	one	PRON
ejpam-6015	163	15	has	have	VERB
ejpam-6015	163	16	+	+	ADJ
ejpam-6015	163	17	∑	∑	PROPN
ejpam-6015	163	18	n=0	n=0	PROPN
ejpam-6015	163	19	∞	∞	PROPN
ejpam-6015	163	20	(	(	PUNCT
ejpam-6015	163	21	(	(	PUNCT
ejpam-6015	163	22	1	1	NUM
ejpam-6015	163	23	+	+	CCONJ
ejpam-6015	163	24	k	k	PROPN
ejpam-6015	163	25	2	2	X
ejpam-6015	163	26	)	)	PUNCT
ejpam-6015	163	27	1	1	NUM
ejpam-6015	163	28	τ	τ	X
ejpam-6015	163	29	)	)	PUNCT
ejpam-6015	163	30	n	n	CCONJ
ejpam-6015	163	31	<	<	X
ejpam-6015	164	1	+	+	PUNCT
ejpam-6015	164	2	∞.	∞.	PROPN
ejpam-6015	164	3	so	so	ADV
ejpam-6015	164	4	,	,	PUNCT
ejpam-6015	164	5	for	for	ADP
ejpam-6015	164	6	all	all	DET
ejpam-6015	164	7	p	p	PRON
ejpam-6015	164	8	≥	≥	NOUN
ejpam-6015	164	9	0	0	NUM
ejpam-6015	164	10	,	,	PUNCT
ejpam-6015	164	11	we	we	PRON
ejpam-6015	164	12	have	have	AUX
ejpam-6015	164	13	d(ςn	d(ςn	NOUN
ejpam-6015	164	14	,	,	PUNCT
ejpam-6015	164	15	ςn+p	ςn+p	PROPN
ejpam-6015	164	16	)	)	PUNCT
ejpam-6015	164	17	≤	≤	NOUN
ejpam-6015	164	18	d(ςn	d(ςn	NOUN
ejpam-6015	164	19	,	,	PUNCT
ejpam-6015	164	20	ςn+1	ςn+1	NUM
ejpam-6015	164	21	)	)	PUNCT
ejpam-6015	164	22	+	+	X
ejpam-6015	164	23	d(ςn+1	d(ςn+1	ADJ
ejpam-6015	164	24	,	,	PUNCT
ejpam-6015	164	25	ςn+2	ςn+2	NUM
ejpam-6015	164	26	)	)	PUNCT
ejpam-6015	164	27	+	+	CCONJ
ejpam-6015	164	28	·	·	PUNCT
ejpam-6015	164	29	·	·	PUNCT
ejpam-6015	164	30	·	·	PUNCT
ejpam-6015	164	31	+	+	CCONJ
ejpam-6015	164	32	d(ςn+p−1	d(ςn+p−1	ADJ
ejpam-6015	164	33	,	,	PUNCT
ejpam-6015	164	34	ςn+p	ςn+p	PROPN
ejpam-6015	164	35	)	)	PUNCT
ejpam-6015	164	36	.	.	PUNCT
ejpam-6015	165	1	that	that	PRON
ejpam-6015	165	2	is	be	AUX
ejpam-6015	165	3	,	,	PUNCT
ejpam-6015	165	4	d(ςn	d(ςn	NOUN
ejpam-6015	165	5	,	,	PUNCT
ejpam-6015	165	6	ςn+p	ςn+p	PROPN
ejpam-6015	165	7	)	)	PUNCT
ejpam-6015	165	8	≤	≤	NOUN
ejpam-6015	165	9	(	(	PUNCT
ejpam-6015	165	10	dθ(ς0	dθ(ς0	NOUN
ejpam-6015	165	11	,	,	PUNCT
ejpam-6015	165	12	ς1	ς1	NOUN
ejpam-6015	165	13	)	)	PUNCT
ejpam-6015	165	14	c	c	NOUN
ejpam-6015	165	15	)	)	PUNCT
ejpam-6015	165	16	1	1	NUM
ejpam-6015	165	17	τ	τ	PROPN
ejpam-6015	165	18	n+p−1∑	n+p−1∑	PROPN
ejpam-6015	165	19	i	i	PROPN
ejpam-6015	165	20	=	=	PROPN
ejpam-6015	165	21	n	n	X
ejpam-6015	165	22	(	(	PUNCT
ejpam-6015	165	23	(	(	PUNCT
ejpam-6015	165	24	1	1	NUM
ejpam-6015	165	25	+	+	CCONJ
ejpam-6015	165	26	k	k	PROPN
ejpam-6015	165	27	2	2	X
ejpam-6015	165	28	)	)	PUNCT
ejpam-6015	165	29	1	1	NUM
ejpam-6015	165	30	τ	τ	NOUN
ejpam-6015	165	31	)	)	PUNCT
ejpam-6015	165	32	n.	n.	NOUN
ejpam-6015	165	33	by	by	ADP
ejpam-6015	165	34	summing	sum	VERB
ejpam-6015	165	35	the	the	DET
ejpam-6015	165	36	geometric	geometric	ADJ
ejpam-6015	165	37	series	series	NOUN
ejpam-6015	165	38	,	,	PUNCT
ejpam-6015	165	39	we	we	PRON
ejpam-6015	165	40	find	find	VERB
ejpam-6015	165	41	+	+	NOUN
ejpam-6015	165	42	∞∑	∞∑	NUM
ejpam-6015	165	43	i	i	NOUN
ejpam-6015	165	44	=	=	NOUN
ejpam-6015	165	45	n	n	X
ejpam-6015	165	46	(	(	PUNCT
ejpam-6015	165	47	(	(	PUNCT
ejpam-6015	165	48	1	1	NUM
ejpam-6015	165	49	+	+	CCONJ
ejpam-6015	165	50	k	k	PROPN
ejpam-6015	165	51	2	2	X
ejpam-6015	165	52	)	)	PUNCT
ejpam-6015	165	53	1	1	NUM
ejpam-6015	165	54	τ	τ	X
ejpam-6015	165	55	)	)	PUNCT
ejpam-6015	165	56	i	i	PRON
ejpam-6015	165	57	→	→	SYM
ejpam-6015	165	58	0	0	PUNCT
ejpam-6015	165	59	as	as	ADP
ejpam-6015	165	60	n→	n→	ADV
ejpam-6015	165	61	+	+	PROPN
ejpam-6015	165	62	∞.	∞.	PROPN
ejpam-6015	165	63	(	(	PUNCT
ejpam-6015	165	64	3.4	3.4	NUM
ejpam-6015	165	65	)	)	PUNCT
ejpam-6015	165	66	the	the	DET
ejpam-6015	165	67	symmetry	symmetry	NOUN
ejpam-6015	165	68	of	of	ADP
ejpam-6015	165	69	d	d	PROPN
ejpam-6015	165	70	leads	lead	VERB
ejpam-6015	165	71	to	to	ADP
ejpam-6015	165	72	lim	lim	PROPN
ejpam-6015	165	73	n	n	CCONJ
ejpam-6015	165	74	,	,	PUNCT
ejpam-6015	165	75	m→+∞	m→+∞	PROPN
ejpam-6015	165	76	d(ςn	d(ςn	NOUN
ejpam-6015	165	77	,	,	PUNCT
ejpam-6015	165	78	ςm	ςm	NOUN
ejpam-6015	165	79	)	)	PUNCT
ejpam-6015	165	80	=	=	SYM
ejpam-6015	166	1	0	0	X
ejpam-6015	166	2	.	.	PUNCT
ejpam-6015	167	1	(	(	PUNCT
ejpam-6015	167	2	3.5	3.5	NUM
ejpam-6015	167	3	)	)	PUNCT
ejpam-6015	167	4	this	this	PRON
ejpam-6015	167	5	implies	imply	VERB
ejpam-6015	167	6	that	that	SCONJ
ejpam-6015	167	7	{	{	PUNCT
ejpam-6015	167	8	ςn	ςn	NOUN
ejpam-6015	167	9	}	}	PUNCT
ejpam-6015	167	10	is	be	AUX
ejpam-6015	167	11	cauchy	cauchy	ADJ
ejpam-6015	167	12	in	in	ADP
ejpam-6015	167	13	the	the	DET
ejpam-6015	167	14	complete	complete	ADJ
ejpam-6015	167	15	ms	ms	NOUN
ejpam-6015	167	16	(	(	PUNCT
ejpam-6015	167	17	x	x	NOUN
ejpam-6015	167	18	,	,	PUNCT
ejpam-6015	167	19	d	d	NOUN
ejpam-6015	167	20	)	)	PUNCT
ejpam-6015	167	21	.	.	PUNCT
ejpam-6015	168	1	so	so	ADV
ejpam-6015	168	2	{	{	PUNCT
ejpam-6015	168	3	ςn	ςn	NOUN
ejpam-6015	168	4	}	}	PUNCT
ejpam-6015	168	5	converges	converge	NOUN
ejpam-6015	168	6	to	to	ADP
ejpam-6015	168	7	some	some	DET
ejpam-6015	168	8	ς∗	ς∗	NOUN
ejpam-6015	168	9	∈	∈	PROPN
ejpam-6015	168	10	x.	x.	NOUN
ejpam-6015	169	1	next	next	ADV
ejpam-6015	169	2	,	,	PUNCT
ejpam-6015	169	3	the	the	DET
ejpam-6015	169	4	lower	low	ADJ
ejpam-6015	169	5	semi	semi	NOUN
ejpam-6015	169	6	-	-	NOUN
ejpam-6015	169	7	continuity	continuity	NOUN
ejpam-6015	169	8	of	of	ADP
ejpam-6015	169	9	ft	ft	NOUN
ejpam-6015	169	10	yields	yield	NOUN
ejpam-6015	169	11	that	that	PRON
ejpam-6015	169	12	d(x∗	d(x∗	NOUN
ejpam-6015	169	13	,	,	PUNCT
ejpam-6015	169	14	t	t	PROPN
ejpam-6015	169	15	ς∗	ς∗	PROPN
ejpam-6015	169	16	)	)	PUNCT
ejpam-6015	170	1	=	=	SYM
ejpam-6015	170	2	ft	ft	PROPN
ejpam-6015	170	3	(	(	PUNCT
ejpam-6015	170	4	ς	ς	PROPN
ejpam-6015	170	5	∗	∗	NOUN
ejpam-6015	170	6	)	)	PUNCT
ejpam-6015	170	7	≤	≤	PROPN
ejpam-6015	170	8	lim	lim	PROPN
ejpam-6015	170	9	inf	inf	PROPN
ejpam-6015	170	10	n→+∞	n→+∞	PROPN
ejpam-6015	170	11	d(ςn	d(ςn	PROPN
ejpam-6015	170	12	,	,	PUNCT
ejpam-6015	170	13	t	t	PROPN
ejpam-6015	170	14	ςn	ςn	NOUN
ejpam-6015	170	15	)	)	PUNCT
ejpam-6015	170	16	≤	≤	NOUN
ejpam-6015	170	17	lim	lim	PROPN
ejpam-6015	170	18	inf	inf	PROPN
ejpam-6015	170	19	n→+∞	n→+∞	PROPN
ejpam-6015	170	20	d(ςn	d(ςn	PROPN
ejpam-6015	170	21	,	,	PUNCT
ejpam-6015	170	22	ςn+1	ςn+1	NUM
ejpam-6015	170	23	)	)	PUNCT
ejpam-6015	170	24	=	=	SYM
ejpam-6015	171	1	0	0	X
ejpam-6015	171	2	.	.	PUNCT
ejpam-6015	172	1	finally	finally	ADV
ejpam-6015	172	2	,	,	PUNCT
ejpam-6015	172	3	we	we	PRON
ejpam-6015	172	4	get	get	VERB
ejpam-6015	172	5	d(ς∗	d(ς∗	ADP
ejpam-6015	172	6	,	,	PUNCT
ejpam-6015	172	7	t	t	PROPN
ejpam-6015	172	8	ς∗	ς∗	PROPN
ejpam-6015	172	9	)	)	PUNCT
ejpam-6015	172	10	=	=	SYM
ejpam-6015	173	1	0	0	NUM
ejpam-6015	173	2	,	,	PUNCT
ejpam-6015	173	3	that	that	ADV
ejpam-6015	173	4	is	is	ADV
ejpam-6015	173	5	,	,	PUNCT
ejpam-6015	173	6	ς∗	ς∗	PROPN
ejpam-6015	173	7	∈	∈	NOUN
ejpam-6015	173	8	tς∗	tς∗	NOUN
ejpam-6015	174	1	=	=	SYM
ejpam-6015	174	2	tς∗.	tς∗.	NOUN
ejpam-6015	174	3	then	then	ADV
ejpam-6015	174	4	,	,	PUNCT
ejpam-6015	174	5	ς∗	ς∗	PROPN
ejpam-6015	174	6	is	be	AUX
ejpam-6015	174	7	a	a	DET
ejpam-6015	174	8	fp	fp	X
ejpam-6015	174	9	of	of	ADP
ejpam-6015	174	10	t.	t.	PROPN
ejpam-6015	174	11	h.	h.	PROPN
ejpam-6015	174	12	aydi	aydi	VERB
ejpam-6015	174	13	et	et	PROPN
ejpam-6015	174	14	al	al	PROPN
ejpam-6015	174	15	.	.	PUNCT
ejpam-6015	174	16	/	/	SYM
ejpam-6015	174	17	eur	eur	PROPN
ejpam-6015	174	18	.	.	PUNCT
ejpam-6015	175	1	j.	j.	PROPN
ejpam-6015	175	2	pure	pure	PROPN
ejpam-6015	175	3	appl	appl	PROPN
ejpam-6015	175	4	.	.	PROPN
ejpam-6015	175	5	math	math	PROPN
ejpam-6015	175	6	,	,	PUNCT
ejpam-6015	175	7	18	18	NUM
ejpam-6015	175	8	(	(	PUNCT
ejpam-6015	175	9	2	2	NUM
ejpam-6015	175	10	)	)	PUNCT
ejpam-6015	175	11	(	(	PUNCT
ejpam-6015	175	12	2025	2025	NUM
ejpam-6015	175	13	)	)	PUNCT
ejpam-6015	175	14	,	,	PUNCT
ejpam-6015	175	15	6015	6015	NUM
ejpam-6015	175	16	8	8	NUM
ejpam-6015	175	17	of	of	ADP
ejpam-6015	175	18	14	14	NUM
ejpam-6015	175	19	3.2	3.2	NUM
ejpam-6015	175	20	.	.	PUNCT
ejpam-6015	176	1	θ−hyperbolic	θ−hyperbolic	ADJ
ejpam-6015	176	2	contractions	contraction	NOUN
ejpam-6015	176	3	via	via	ADP
ejpam-6015	176	4	manageable	manageable	ADJ
ejpam-6015	176	5	functions	function	NOUN
ejpam-6015	176	6	in	in	ADP
ejpam-6015	176	7	2014	2014	NUM
ejpam-6015	176	8	,	,	PUNCT
ejpam-6015	176	9	a	a	DET
ejpam-6015	176	10	new	new	ADJ
ejpam-6015	176	11	class	class	NOUN
ejpam-6015	176	12	of	of	ADP
ejpam-6015	176	13	mappings	mapping	NOUN
ejpam-6015	176	14	called	call	VERB
ejpam-6015	176	15	manageable	manageable	ADJ
ejpam-6015	176	16	functions	function	NOUN
ejpam-6015	176	17	was	be	AUX
ejpam-6015	176	18	explored	explore	VERB
ejpam-6015	176	19	by	by	ADP
ejpam-6015	176	20	du	du	PROPN
ejpam-6015	176	21	and	and	CCONJ
ejpam-6015	176	22	khojasteh	khojasteh	ADJ
ejpam-6015	177	1	[	[	X
ejpam-6015	177	2	14	14	NUM
ejpam-6015	177	3	]	]	PUNCT
ejpam-6015	177	4	.	.	PUNCT
ejpam-6015	178	1	they	they	PRON
ejpam-6015	178	2	used	use	VERB
ejpam-6015	178	3	this	this	DET
ejpam-6015	178	4	class	class	NOUN
ejpam-6015	178	5	to	to	PART
ejpam-6015	178	6	obtain	obtain	VERB
ejpam-6015	178	7	some	some	DET
ejpam-6015	178	8	fp	fp	NOUN
ejpam-6015	178	9	theorems	theorem	NOUN
ejpam-6015	178	10	.	.	PUNCT
ejpam-6015	179	1	in	in	ADP
ejpam-6015	179	2	2017	2017	NUM
ejpam-6015	179	3	,	,	PUNCT
ejpam-6015	179	4	hussain	hussain	PROPN
ejpam-6015	179	5	et	et	PROPN
ejpam-6015	179	6	al	al	PROPN
ejpam-6015	179	7	.	.	PUNCT
ejpam-6015	180	1	[	[	X
ejpam-6015	180	2	15	15	NUM
ejpam-6015	180	3	]	]	PUNCT
ejpam-6015	180	4	established	establish	VERB
ejpam-6015	180	5	some	some	DET
ejpam-6015	180	6	fp	fp	NOUN
ejpam-6015	180	7	theorems	theorem	NOUN
ejpam-6015	180	8	in	in	ADP
ejpam-6015	180	9	the	the	DET
ejpam-6015	180	10	setting	setting	NOUN
ejpam-6015	180	11	of	of	ADP
ejpam-6015	180	12	mss	mss	NOUN
ejpam-6015	180	13	for	for	ADP
ejpam-6015	180	14	contraction	contraction	NOUN
ejpam-6015	180	15	mappings	mapping	NOUN
ejpam-6015	180	16	via	via	ADP
ejpam-6015	180	17	manageable	manageable	ADJ
ejpam-6015	180	18	functions	function	NOUN
ejpam-6015	180	19	.	.	PUNCT
ejpam-6015	181	1	definition	definition	NOUN
ejpam-6015	181	2	6	6	NUM
ejpam-6015	181	3	.	.	PUNCT
ejpam-6015	182	1	[	[	X
ejpam-6015	182	2	14	14	NUM
ejpam-6015	182	3	]	]	X
ejpam-6015	182	4	a	a	DET
ejpam-6015	182	5	manageable	manageable	ADJ
ejpam-6015	182	6	function	function	NOUN
ejpam-6015	182	7	η	η	PROPN
ejpam-6015	182	8	:	:	PUNCT
ejpam-6015	182	9	r×	r×	NOUN
ejpam-6015	182	10	r	r	NOUN
ejpam-6015	182	11	→	→	SYM
ejpam-6015	182	12	r	r	NOUN
ejpam-6015	182	13	is	be	AUX
ejpam-6015	182	14	a	a	DET
ejpam-6015	182	15	function	function	NOUN
ejpam-6015	182	16	so	so	SCONJ
ejpam-6015	182	17	that	that	SCONJ
ejpam-6015	182	18	:	:	PUNCT
ejpam-6015	182	19	(	(	PUNCT
ejpam-6015	182	20	η1	η1	NOUN
ejpam-6015	182	21	)	)	PUNCT
ejpam-6015	182	22	η(ℏ	η(ℏ	PROPN
ejpam-6015	182	23	,	,	PUNCT
ejpam-6015	182	24	ℓ	ℓ	NUM
ejpam-6015	182	25	)	)	PUNCT
ejpam-6015	182	26	<	<	X
ejpam-6015	182	27	ℓ−	ℓ−	ADP
ejpam-6015	182	28	ℏ	ℏ	PROPN
ejpam-6015	182	29	for	for	ADP
ejpam-6015	182	30	all	all	DET
ejpam-6015	182	31	ℏ	ℏ	PROPN
ejpam-6015	182	32	,	,	PUNCT
ejpam-6015	182	33	ℓ	ℓ	PROPN
ejpam-6015	182	34	>	>	X
ejpam-6015	182	35	0	0	NUM
ejpam-6015	182	36	;	;	PUNCT
ejpam-6015	182	37	(	(	PUNCT
ejpam-6015	182	38	η2	η2	X
ejpam-6015	182	39	)	)	PUNCT
ejpam-6015	182	40	for	for	ADP
ejpam-6015	182	41	each	each	DET
ejpam-6015	182	42	bounded	bound	VERB
ejpam-6015	182	43	{	{	PUNCT
ejpam-6015	182	44	ℏn	ℏn	NOUN
ejpam-6015	182	45	}	}	PUNCT
ejpam-6015	182	46	in	in	ADP
ejpam-6015	182	47	(	(	PUNCT
ejpam-6015	182	48	0,+∞	0,+∞	NUM
ejpam-6015	182	49	)	)	PUNCT
ejpam-6015	182	50	and	and	CCONJ
ejpam-6015	182	51	each	each	DET
ejpam-6015	182	52	non	non	ADJ
ejpam-6015	182	53	-	-	ADJ
ejpam-6015	182	54	increasing	increase	VERB
ejpam-6015	182	55	{	{	PUNCT
ejpam-6015	182	56	ℓn	ℓn	NOUN
ejpam-6015	182	57	}	}	PUNCT
ejpam-6015	182	58	in	in	ADP
ejpam-6015	182	59	(	(	PUNCT
ejpam-6015	182	60	0,+∞	0,+∞	NUM
ejpam-6015	182	61	)	)	PUNCT
ejpam-6015	182	62	,	,	PUNCT
ejpam-6015	182	63	lim	lim	PROPN
ejpam-6015	182	64	sup	sup	PROPN
ejpam-6015	182	65	n→+∞	n→+∞	VERB
ejpam-6015	182	66	ℏn	ℏn	NOUN
ejpam-6015	182	67	+	+	SYM
ejpam-6015	182	68	η(ℏn	η(ℏn	NOUN
ejpam-6015	182	69	,	,	PUNCT
ejpam-6015	182	70	ℓn	ℓn	ADV
ejpam-6015	182	71	)	)	PUNCT
ejpam-6015	182	72	ℓn	ℓn	ADV
ejpam-6015	182	73	<	<	X
ejpam-6015	183	1	1	1	X
ejpam-6015	183	2	.	.	PUNCT
ejpam-6015	183	3	let	let	VERB
ejpam-6015	183	4	m̂an(r	m̂an(r	NOUN
ejpam-6015	183	5	)	)	PUNCT
ejpam-6015	183	6	be	be	AUX
ejpam-6015	183	7	the	the	DET
ejpam-6015	183	8	set	set	NOUN
ejpam-6015	183	9	of	of	ADP
ejpam-6015	183	10	manageable	manageable	ADJ
ejpam-6015	183	11	functions	function	NOUN
ejpam-6015	183	12	.	.	PUNCT
ejpam-6015	184	1	we	we	PRON
ejpam-6015	184	2	give	give	VERB
ejpam-6015	184	3	the	the	DET
ejpam-6015	184	4	next	next	ADJ
ejpam-6015	184	5	two	two	NUM
ejpam-6015	184	6	examples	example	NOUN
ejpam-6015	184	7	.	.	PUNCT
ejpam-6015	185	1	example	example	NOUN
ejpam-6015	186	1	3	3	NUM
ejpam-6015	186	2	.	.	PUNCT
ejpam-6015	187	1	[	[	X
ejpam-6015	187	2	14	14	NUM
ejpam-6015	187	3	]	]	PUNCT
ejpam-6015	187	4	let	let	VERB
ejpam-6015	187	5	℘	℘	PROPN
ejpam-6015	187	6	∈	∈	PROPN
ejpam-6015	188	1	[	[	X
ejpam-6015	188	2	0	0	NUM
ejpam-6015	188	3	,	,	PUNCT
ejpam-6015	188	4	1	1	NUM
ejpam-6015	188	5	)	)	PUNCT
ejpam-6015	188	6	.	.	PUNCT
ejpam-6015	189	1	then	then	ADV
ejpam-6015	189	2	ηk	ηk	ADV
ejpam-6015	189	3	:	:	PUNCT
ejpam-6015	189	4	r2	r2	PROPN
ejpam-6015	189	5	→	→	SYM
ejpam-6015	189	6	r	r	NOUN
ejpam-6015	189	7	defined	define	VERB
ejpam-6015	189	8	by	by	ADP
ejpam-6015	189	9	ηk(ℏ	ηk(ℏ	NOUN
ejpam-6015	189	10	,	,	PUNCT
ejpam-6015	189	11	ℓ	ℓ	X
ejpam-6015	189	12	)	)	PUNCT
ejpam-6015	189	13	=	=	PUNCT
ejpam-6015	190	1	℘ℓ−	℘ℓ−	NOUN
ejpam-6015	190	2	ℏ	ℏ	PROPN
ejpam-6015	190	3	is	be	AUX
ejpam-6015	190	4	a	a	DET
ejpam-6015	190	5	manageable	manageable	ADJ
ejpam-6015	190	6	function	function	NOUN
ejpam-6015	190	7	.	.	PUNCT
ejpam-6015	191	1	example	example	NOUN
ejpam-6015	192	1	4	4	NUM
ejpam-6015	192	2	.	.	PUNCT
ejpam-6015	193	1	[	[	X
ejpam-6015	193	2	16	16	NUM
ejpam-6015	193	3	]	]	X
ejpam-6015	193	4	let	let	VERB
ejpam-6015	193	5	η	η	NOUN
ejpam-6015	193	6	:	:	PUNCT
ejpam-6015	193	7	r×	r×	NOUN
ejpam-6015	193	8	r	r	NOUN
ejpam-6015	193	9	→	→	SYM
ejpam-6015	193	10	r	r	NOUN
ejpam-6015	193	11	be	be	VERB
ejpam-6015	193	12	the	the	DET
ejpam-6015	193	13	function	function	NOUN
ejpam-6015	193	14	defined	define	VERB
ejpam-6015	193	15	by	by	ADP
ejpam-6015	193	16	η(ℏ	η(ℏ	PROPN
ejpam-6015	193	17	,	,	PUNCT
ejpam-6015	193	18	ℓ	ℓ	NUM
ejpam-6015	193	19	)	)	PUNCT
ejpam-6015	193	20	=	=	PRON
ejpam-6015	193	21	{	{	PUNCT
ejpam-6015	193	22	ψ(ℓ)−	ψ(ℓ)−	PROPN
ejpam-6015	193	23	ϕ(ℏ	ϕ(ℏ	PROPN
ejpam-6015	193	24	)	)	PUNCT
ejpam-6015	193	25	if	if	SCONJ
ejpam-6015	193	26	(	(	PUNCT
ejpam-6015	193	27	ℏ	ℏ	PROPN
ejpam-6015	193	28	,	,	PUNCT
ejpam-6015	193	29	ℓ	ℓ	NUM
ejpam-6015	193	30	)	)	PUNCT
ejpam-6015	193	31	∈	∈	PROPN
ejpam-6015	194	1	[	[	X
ejpam-6015	194	2	0,+∞)×	0,+∞)×	NOUN
ejpam-6015	195	1	[	[	X
ejpam-6015	195	2	0,+∞	0,+∞	NUM
ejpam-6015	195	3	)	)	PUNCT
ejpam-6015	195	4	,	,	PUNCT
ejpam-6015	195	5	f(ℓ	f(ℓ	PROPN
ejpam-6015	195	6	,	,	PUNCT
ejpam-6015	195	7	ℏ	ℏ	NOUN
ejpam-6015	195	8	)	)	PUNCT
ejpam-6015	195	9	otherwise	otherwise	ADV
ejpam-6015	195	10	,	,	PUNCT
ejpam-6015	195	11	where	where	SCONJ
ejpam-6015	195	12	f	f	X
ejpam-6015	195	13	:	:	PUNCT
ejpam-6015	195	14	r2	r2	PROPN
ejpam-6015	195	15	→	→	PUNCT
ejpam-6015	195	16	r	r	NOUN
ejpam-6015	195	17	is	be	AUX
ejpam-6015	195	18	a	a	DET
ejpam-6015	195	19	given	give	VERB
ejpam-6015	195	20	function	function	NOUN
ejpam-6015	195	21	and	and	CCONJ
ejpam-6015	195	22	ψ	ψ	NOUN
ejpam-6015	195	23	,	,	PUNCT
ejpam-6015	195	24	ϕ	ϕ	X
ejpam-6015	195	25	:	:	PUNCT
ejpam-6015	196	1	[	[	X
ejpam-6015	196	2	0,+∞	0,+∞	NUM
ejpam-6015	196	3	)	)	PUNCT
ejpam-6015	196	4	→	→	SYM
ejpam-6015	196	5	r	r	NOUN
ejpam-6015	196	6	are	be	AUX
ejpam-6015	196	7	two	two	NUM
ejpam-6015	196	8	functions	function	NOUN
ejpam-6015	196	9	sp	sp	ADP
ejpam-6015	196	10	that	that	DET
ejpam-6015	196	11	•	•	NUM
ejpam-6015	196	12	ψ(ℏ	ψ(ℏ	NOUN
ejpam-6015	196	13	)	)	PUNCT
ejpam-6015	196	14	<	<	X
ejpam-6015	197	1	ℏ	ℏ	X
ejpam-6015	197	2	≤	≤	PUNCT
ejpam-6015	197	3	ϕ(t	ϕ(t	NUM
ejpam-6015	197	4	)	)	PUNCT
ejpam-6015	197	5	for	for	ADP
ejpam-6015	197	6	all	all	DET
ejpam-6015	197	7	ℏ	ℏ	PROPN
ejpam-6015	197	8	>	>	X
ejpam-6015	197	9	0	0	NUM
ejpam-6015	197	10	,	,	PUNCT
ejpam-6015	197	11	and	and	CCONJ
ejpam-6015	197	12	•	•	NUM
ejpam-6015	197	13	lim	lim	NOUN
ejpam-6015	197	14	supr→ℏ+	supr→ℏ+	VERB
ejpam-6015	197	15	ψ(r	ψ(r	NOUN
ejpam-6015	197	16	)	)	PUNCT
ejpam-6015	197	17	r	r	NOUN
ejpam-6015	197	18	<	<	X
ejpam-6015	197	19	1	1	NUM
ejpam-6015	197	20	for	for	ADP
ejpam-6015	197	21	any	any	DET
ejpam-6015	197	22	t	t	PROPN
ejpam-6015	197	23	≥	≥	NOUN
ejpam-6015	197	24	0	0	NUM
ejpam-6015	197	25	.	.	PUNCT
ejpam-6015	198	1	then	then	ADV
ejpam-6015	198	2	,	,	PUNCT
ejpam-6015	198	3	η	η	PROPN
ejpam-6015	198	4	∈	∈	PROPN
ejpam-6015	198	5	m̂an(r	m̂an(r	PROPN
ejpam-6015	198	6	)	)	PUNCT
ejpam-6015	198	7	.	.	PUNCT
ejpam-6015	199	1	indeed	indeed	ADV
ejpam-6015	199	2	,	,	PUNCT
ejpam-6015	199	3	for	for	ADP
ejpam-6015	199	4	any	any	DET
ejpam-6015	199	5	s	s	NOUN
ejpam-6015	199	6	,	,	PUNCT
ejpam-6015	199	7	t	t	X
ejpam-6015	199	8	>	>	X
ejpam-6015	199	9	0	0	NUM
ejpam-6015	199	10	,	,	PUNCT
ejpam-6015	199	11	η(t	η(t	NOUN
ejpam-6015	199	12	,	,	PUNCT
ejpam-6015	199	13	s	s	PART
ejpam-6015	199	14	)	)	PUNCT
ejpam-6015	199	15	=	=	SYM
ejpam-6015	199	16	ψ(s)−	ψ(s)−	PROPN
ejpam-6015	199	17	ϕ(t	ϕ(t	NUM
ejpam-6015	199	18	)	)	PUNCT
ejpam-6015	199	19	<	<	X
ejpam-6015	199	20	s−	s−	PROPN
ejpam-6015	199	21	t	t	PROPN
ejpam-6015	199	22	,	,	PUNCT
ejpam-6015	199	23	so	so	ADV
ejpam-6015	199	24	,	,	PUNCT
ejpam-6015	199	25	(	(	PUNCT
ejpam-6015	199	26	η1	η1	NOUN
ejpam-6015	199	27	)	)	PUNCT
ejpam-6015	199	28	holds	hold	VERB
ejpam-6015	199	29	.	.	PUNCT
ejpam-6015	200	1	let	let	AUX
ejpam-6015	200	2	{	{	PUNCT
ejpam-6015	200	3	tn	tn	NOUN
ejpam-6015	200	4	}	}	PUNCT
ejpam-6015	200	5	be	be	AUX
ejpam-6015	200	6	a	a	DET
ejpam-6015	200	7	bounded	bounded	ADJ
ejpam-6015	200	8	and	and	CCONJ
ejpam-6015	200	9	{	{	PUNCT
ejpam-6015	200	10	sn	sn	NOUN
ejpam-6015	200	11	}	}	PUNCT
ejpam-6015	200	12	be	be	AUX
ejpam-6015	200	13	a	a	DET
ejpam-6015	200	14	non	non	ADJ
ejpam-6015	200	15	-	-	ADJ
ejpam-6015	200	16	increasing	increase	VERB
ejpam-6015	200	17	in	in	ADP
ejpam-6015	200	18	(	(	PUNCT
ejpam-6015	200	19	0,+∞	0,+∞	NUM
ejpam-6015	200	20	)	)	PUNCT
ejpam-6015	200	21	.	.	PUNCT
ejpam-6015	201	1	then	then	ADV
ejpam-6015	201	2	limn→+∞	limn→+∞	AUX
ejpam-6015	201	3	sn	sn	PROPN
ejpam-6015	201	4	exists	exist	VERB
ejpam-6015	201	5	in	in	ADP
ejpam-6015	201	6	[	[	NOUN
ejpam-6015	201	7	0,+∞	0,+∞	NUM
ejpam-6015	201	8	)	)	PUNCT
ejpam-6015	201	9	.	.	PUNCT
ejpam-6015	202	1	hence	hence	ADV
ejpam-6015	202	2	,	,	PUNCT
ejpam-6015	202	3	lim	lim	PROPN
ejpam-6015	202	4	sup	sup	PROPN
ejpam-6015	202	5	n→+∞	n→+∞	PROPN
ejpam-6015	202	6	ψ(sn	ψ(sn	NUM
ejpam-6015	202	7	)	)	PUNCT
ejpam-6015	202	8	sn	sn	NOUN
ejpam-6015	203	1	=	=	SYM
ejpam-6015	203	2	lim	lim	PROPN
ejpam-6015	203	3	sup	sup	NOUN
ejpam-6015	203	4	r→t+	r→t+	VERB
ejpam-6015	203	5	ψ(r	ψ(r	NOUN
ejpam-6015	203	6	)	)	PUNCT
ejpam-6015	203	7	r	r	NOUN
ejpam-6015	203	8	<	<	X
ejpam-6015	203	9	1	1	NUM
ejpam-6015	203	10	.	.	PUNCT
ejpam-6015	204	1	thus	thus	ADV
ejpam-6015	204	2	,	,	PUNCT
ejpam-6015	204	3	we	we	PRON
ejpam-6015	204	4	get	get	VERB
ejpam-6015	204	5	lim	lim	PROPN
ejpam-6015	204	6	sup	sup	PROPN
ejpam-6015	205	1	n→+∞	n→+∞	PROPN
ejpam-6015	205	2	tn	tn	PROPN
ejpam-6015	205	3	+	+	SYM
ejpam-6015	205	4	η(tn	η(tn	PROPN
ejpam-6015	205	5	,	,	PUNCT
ejpam-6015	205	6	sn	sn	NOUN
ejpam-6015	205	7	)	)	PUNCT
ejpam-6015	205	8	sn	sn	PROPN
ejpam-6015	205	9	=	=	SYM
ejpam-6015	205	10	lim	lim	PROPN
ejpam-6015	205	11	sup	sup	PROPN
ejpam-6015	205	12	n→+∞	n→+∞	PROPN
ejpam-6015	205	13	ψ(sn	ψ(sn	NUM
ejpam-6015	205	14	)	)	PUNCT
ejpam-6015	205	15	+	+	NUM
ejpam-6015	205	16	tn	tn	NOUN
ejpam-6015	205	17	−	−	NOUN
ejpam-6015	205	18	ϕ(tn	ϕ(tn	NOUN
ejpam-6015	205	19	)	)	PUNCT
ejpam-6015	205	20	sn	sn	PROPN
ejpam-6015	205	21	≤	≤	PROPN
ejpam-6015	206	1	lim	lim	PROPN
ejpam-6015	206	2	sup	sup	PROPN
ejpam-6015	206	3	n→+∞	n→+∞	PROPN
ejpam-6015	206	4	ψ(sn	ψ(sn	NUM
ejpam-6015	206	5	)	)	PUNCT
ejpam-6015	206	6	sn	sn	NOUN
ejpam-6015	206	7	<	<	X
ejpam-6015	206	8	1	1	X
ejpam-6015	206	9	.	.	PUNCT
ejpam-6015	207	1	it	it	PRON
ejpam-6015	207	2	follows	follow	VERB
ejpam-6015	207	3	that	that	SCONJ
ejpam-6015	207	4	(	(	PUNCT
ejpam-6015	207	5	η2	η2	X
ejpam-6015	207	6	)	)	PUNCT
ejpam-6015	207	7	holds	hold	NOUN
ejpam-6015	207	8	.	.	PUNCT
ejpam-6015	208	1	h.	h.	PROPN
ejpam-6015	208	2	aydi	aydi	VERB
ejpam-6015	208	3	et	et	PROPN
ejpam-6015	208	4	al	al	PROPN
ejpam-6015	208	5	.	.	PUNCT
ejpam-6015	208	6	/	/	SYM
ejpam-6015	208	7	eur	eur	PROPN
ejpam-6015	208	8	.	.	PUNCT
ejpam-6015	209	1	j.	j.	PROPN
ejpam-6015	209	2	pure	pure	PROPN
ejpam-6015	209	3	appl	appl	PROPN
ejpam-6015	209	4	.	.	PROPN
ejpam-6015	209	5	math	math	PROPN
ejpam-6015	209	6	,	,	PUNCT
ejpam-6015	209	7	18	18	NUM
ejpam-6015	209	8	(	(	PUNCT
ejpam-6015	209	9	2	2	NUM
ejpam-6015	209	10	)	)	PUNCT
ejpam-6015	209	11	(	(	PUNCT
ejpam-6015	209	12	2025	2025	NUM
ejpam-6015	209	13	)	)	PUNCT
ejpam-6015	209	14	,	,	PUNCT
ejpam-6015	209	15	6015	6015	NUM
ejpam-6015	209	16	9	9	NUM
ejpam-6015	209	17	of	of	ADP
ejpam-6015	209	18	14	14	NUM
ejpam-6015	209	19	the	the	DET
ejpam-6015	209	20	next	next	ADJ
ejpam-6015	209	21	lemma	lemma	PROPN
ejpam-6015	209	22	is	be	AUX
ejpam-6015	209	23	needful	needful	ADJ
ejpam-6015	209	24	.	.	PUNCT
ejpam-6015	210	1	lemma	lemma	PROPN
ejpam-6015	210	2	2	2	X
ejpam-6015	210	3	.	.	PUNCT
ejpam-6015	211	1	let	let	VERB
ejpam-6015	211	2	(	(	PUNCT
ejpam-6015	211	3	x	x	NOUN
ejpam-6015	211	4	,	,	PUNCT
ejpam-6015	211	5	d	d	NOUN
ejpam-6015	211	6	)	)	PUNCT
ejpam-6015	211	7	be	be	AUX
ejpam-6015	211	8	a	a	DET
ejpam-6015	211	9	ms	ms	PROPN
ejpam-6015	211	10	,	,	PUNCT
ejpam-6015	211	11	ξ	ξ	X
ejpam-6015	211	12	∈	∈	PROPN
ejpam-6015	211	13	cb(x	cb(x	NUM
ejpam-6015	211	14	)	)	PUNCT
ejpam-6015	211	15	and	and	CCONJ
ejpam-6015	211	16	k	k	X
ejpam-6015	211	17	>	>	X
ejpam-6015	211	18	0	0	X
ejpam-6015	211	19	.	.	PUNCT
ejpam-6015	212	1	let	let	VERB
ejpam-6015	212	2	θ	θ	PROPN
ejpam-6015	212	3	∈	∈	PROPN
ejpam-6015	212	4	θτ	θτ	NOUN
ejpam-6015	212	5	for	for	ADP
ejpam-6015	212	6	some	some	DET
ejpam-6015	212	7	τ	τ	PROPN
ejpam-6015	212	8	>	>	X
ejpam-6015	212	9	0	0	PROPN
ejpam-6015	212	10	.	.	PUNCT
ejpam-6015	213	1	when	when	SCONJ
ejpam-6015	213	2	a	a	DET
ejpam-6015	213	3	∈	∈	PROPN
ejpam-6015	213	4	x	x	X
ejpam-6015	213	5	and	and	CCONJ
ejpam-6015	213	6	dθ(a	dθ(a	NOUN
ejpam-6015	213	7	,	,	PUNCT
ejpam-6015	213	8	ξ	ξ	X
ejpam-6015	213	9	)	)	PUNCT
ejpam-6015	213	10	<	<	X
ejpam-6015	213	11	k	k	X
ejpam-6015	213	12	,	,	PUNCT
ejpam-6015	213	13	then	then	ADV
ejpam-6015	213	14	there	there	PRON
ejpam-6015	213	15	is	be	VERB
ejpam-6015	213	16	b	b	PROPN
ejpam-6015	213	17	∈	∈	ADP
ejpam-6015	213	18	ξ	ξ	NOUN
ejpam-6015	214	1	so	so	ADV
ejpam-6015	214	2	that	that	PRON
ejpam-6015	214	3	dθ(a	dθ(a	NOUN
ejpam-6015	214	4	,	,	PUNCT
ejpam-6015	214	5	b	b	X
ejpam-6015	214	6	)	)	PUNCT
ejpam-6015	214	7	<	<	X
ejpam-6015	214	8	k.	k.	PROPN
ejpam-6015	214	9	proof	proof	PROPN
ejpam-6015	214	10	.	.	PUNCT
ejpam-6015	215	1	let	let	VERB
ejpam-6015	215	2	a	a	DET
ejpam-6015	215	3	∈	∈	NOUN
ejpam-6015	215	4	x	x	X
ejpam-6015	215	5	and	and	CCONJ
ejpam-6015	215	6	suppose	suppose	VERB
ejpam-6015	215	7	that	that	SCONJ
ejpam-6015	215	8	dθ(a	dθ(a	NOUN
ejpam-6015	215	9	,	,	PUNCT
ejpam-6015	215	10	b	b	NOUN
ejpam-6015	215	11	)	)	PUNCT
ejpam-6015	215	12	<	<	X
ejpam-6015	215	13	k.	k.	PROPN
ejpam-6015	215	14	recall	recall	VERB
ejpam-6015	215	15	that	that	DET
ejpam-6015	215	16	dθ(a	dθ(a	NOUN
ejpam-6015	215	17	,	,	PUNCT
ejpam-6015	215	18	b	b	NOUN
ejpam-6015	215	19	)	)	PUNCT
ejpam-6015	215	20	:	:	PUNCT
ejpam-6015	216	1	=	=	NUM
ejpam-6015	216	2	inf	inf	PROPN
ejpam-6015	216	3	b∈b	b∈b	NOUN
ejpam-6015	216	4	dθ(a	dθ(a	PUNCT
ejpam-6015	216	5	,	,	PUNCT
ejpam-6015	216	6	b	b	NOUN
ejpam-6015	216	7	)	)	PUNCT
ejpam-6015	216	8	.	.	PUNCT
ejpam-6015	217	1	suppose	suppose	VERB
ejpam-6015	217	2	in	in	ADP
ejpam-6015	217	3	the	the	DET
ejpam-6015	217	4	contrary	contrary	NOUN
ejpam-6015	217	5	,	,	PUNCT
ejpam-6015	217	6	for	for	ADP
ejpam-6015	217	7	all	all	DET
ejpam-6015	217	8	b	b	NOUN
ejpam-6015	217	9	∈	∈	ADP
ejpam-6015	217	10	b	b	NOUN
ejpam-6015	217	11	we	we	PRON
ejpam-6015	217	12	have	have	VERB
ejpam-6015	217	13	dθ(a	dθ(a	PUNCT
ejpam-6015	217	14	,	,	PUNCT
ejpam-6015	217	15	b	b	X
ejpam-6015	217	16	)	)	PUNCT
ejpam-6015	217	17	≥	≥	NOUN
ejpam-6015	217	18	k.	k.	PUNCT
ejpam-6015	218	1	thus	thus	ADV
ejpam-6015	218	2	,	,	PUNCT
ejpam-6015	218	3	infb∈b	infb∈b	ADJ
ejpam-6015	218	4	dθ(a	dθ(a	NOUN
ejpam-6015	218	5	,	,	PUNCT
ejpam-6015	218	6	b	b	NOUN
ejpam-6015	218	7	)	)	PUNCT
ejpam-6015	218	8	=	=	PUNCT
ejpam-6015	218	9	dθ(a	dθ(a	PUNCT
ejpam-6015	218	10	,	,	PUNCT
ejpam-6015	218	11	b	b	NOUN
ejpam-6015	218	12	)	)	PUNCT
ejpam-6015	218	13	≥	≥	NOUN
ejpam-6015	219	1	k	k	NOUN
ejpam-6015	219	2	,	,	PUNCT
ejpam-6015	219	3	which	which	PRON
ejpam-6015	219	4	is	be	AUX
ejpam-6015	219	5	impossible	impossible	ADJ
ejpam-6015	219	6	.	.	PUNCT
ejpam-6015	220	1	so	so	ADV
ejpam-6015	220	2	there	there	PRON
ejpam-6015	220	3	exists	exist	VERB
ejpam-6015	220	4	a	a	DET
ejpam-6015	220	5	point	point	NOUN
ejpam-6015	220	6	b	b	NOUN
ejpam-6015	220	7	∈	∈	NOUN
ejpam-6015	220	8	b	b	NOUN
ejpam-6015	220	9	such	such	ADJ
ejpam-6015	220	10	that	that	DET
ejpam-6015	220	11	dθ(a	dθ(a	NOUN
ejpam-6015	220	12	,	,	PUNCT
ejpam-6015	220	13	b	b	X
ejpam-6015	220	14	)	)	PUNCT
ejpam-6015	220	15	<	<	X
ejpam-6015	220	16	k.	k.	X
ejpam-6015	221	1	our	our	PRON
ejpam-6015	221	2	second	second	ADJ
ejpam-6015	221	3	result	result	NOUN
ejpam-6015	221	4	for	for	ADP
ejpam-6015	221	5	multivalued	multivalued	ADJ
ejpam-6015	221	6	mappings	mapping	NOUN
ejpam-6015	221	7	involves	involve	VERB
ejpam-6015	221	8	manageable	manageable	ADJ
ejpam-6015	221	9	functions	function	NOUN
ejpam-6015	221	10	.	.	PUNCT
ejpam-6015	222	1	theorem	theorem	NOUN
ejpam-6015	222	2	4	4	NUM
ejpam-6015	222	3	.	.	PUNCT
ejpam-6015	223	1	let	let	AUX
ejpam-6015	223	2	(	(	PUNCT
ejpam-6015	223	3	x	x	NOUN
ejpam-6015	223	4	,	,	PUNCT
ejpam-6015	223	5	d	d	NOUN
ejpam-6015	223	6	)	)	PUNCT
ejpam-6015	223	7	be	be	AUX
ejpam-6015	223	8	a	a	DET
ejpam-6015	223	9	complete	complete	ADJ
ejpam-6015	223	10	ms	ms	NOUN
ejpam-6015	223	11	and	and	CCONJ
ejpam-6015	223	12	t	t	PROPN
ejpam-6015	223	13	:	:	PUNCT
ejpam-6015	223	14	x	x	X
ejpam-6015	223	15	→	→	X
ejpam-6015	223	16	cb(x	cb(x	NUM
ejpam-6015	223	17	)	)	PUNCT
ejpam-6015	223	18	.	.	PUNCT
ejpam-6015	224	1	let	let	VERB
ejpam-6015	224	2	θ	θ	PROPN
ejpam-6015	224	3	∈	∈	PROPN
ejpam-6015	224	4	θτ	θτ	NOUN
ejpam-6015	224	5	for	for	ADP
ejpam-6015	224	6	some	some	DET
ejpam-6015	224	7	τ	τ	PROPN
ejpam-6015	224	8	>	>	X
ejpam-6015	224	9	0	0	X
ejpam-6015	224	10	.	.	PUNCT
ejpam-6015	224	11	suppose	suppose	VERB
ejpam-6015	224	12	there	there	PRON
ejpam-6015	224	13	is	be	VERB
ejpam-6015	224	14	η	η	PROPN
ejpam-6015	224	15	∈	∈	PROPN
ejpam-6015	224	16	m̂an(r	m̂an(r	NOUN
ejpam-6015	224	17	)	)	PUNCT
ejpam-6015	224	18	so	so	SCONJ
ejpam-6015	224	19	that	that	PRON
ejpam-6015	224	20	η(hθ(tx	η(hθ(tx	NOUN
ejpam-6015	224	21	,	,	PUNCT
ejpam-6015	224	22	ty	ty	NOUN
ejpam-6015	224	23	)	)	PUNCT
ejpam-6015	224	24	,	,	PUNCT
ejpam-6015	224	25	dθ(ϖ	dθ(ϖ	PROPN
ejpam-6015	224	26	,	,	PUNCT
ejpam-6015	224	27	ς	ς	NOUN
ejpam-6015	224	28	)	)	PUNCT
ejpam-6015	224	29	)	)	PUNCT
ejpam-6015	224	30	≥	≥	NOUN
ejpam-6015	224	31	0	0	NUM
ejpam-6015	224	32	∀ϖ	∀ϖ	ADJ
ejpam-6015	224	33	,	,	PUNCT
ejpam-6015	224	34	ς	ς	PROPN
ejpam-6015	224	35	∈	∈	PROPN
ejpam-6015	224	36	x.	x.	NOUN
ejpam-6015	224	37	(	(	PUNCT
ejpam-6015	224	38	6	6	NUM
ejpam-6015	224	39	)	)	PUNCT
ejpam-6015	224	40	if	if	SCONJ
ejpam-6015	224	41	ft	ft	NOUN
ejpam-6015	224	42	is	be	AUX
ejpam-6015	224	43	lower	low	ADJ
ejpam-6015	224	44	semi	semi	ADJ
ejpam-6015	224	45	-	-	ADJ
ejpam-6015	224	46	continuous	continuous	ADJ
ejpam-6015	224	47	,	,	PUNCT
ejpam-6015	224	48	then	then	ADV
ejpam-6015	224	49	t	t	PROPN
ejpam-6015	224	50	admits	admit	VERB
ejpam-6015	224	51	a	a	DET
ejpam-6015	224	52	fp	fp	NOUN
ejpam-6015	224	53	.	.	NOUN
ejpam-6015	224	54	proof	proof	NOUN
ejpam-6015	224	55	.	.	PUNCT
ejpam-6015	225	1	let	let	VERB
ejpam-6015	225	2	ς0	ς0	PROPN
ejpam-6015	225	3	∈	∈	NOUN
ejpam-6015	225	4	x	x	X
ejpam-6015	225	5	and	and	CCONJ
ejpam-6015	225	6	ς1	ς1	PROPN
ejpam-6015	225	7	∈	∈	PROPN
ejpam-6015	225	8	tς0	tς0	PROPN
ejpam-6015	225	9	.	.	PUNCT
ejpam-6015	226	1	when	when	SCONJ
ejpam-6015	226	2	ς1	ς1	NOUN
ejpam-6015	226	3	=	=	SYM
ejpam-6015	226	4	ς0	ς0	PROPN
ejpam-6015	226	5	or	or	CCONJ
ejpam-6015	226	6	ς1	ς1	PROPN
ejpam-6015	226	7	∈	∈	PROPN
ejpam-6015	226	8	tς1	tς1	PROPN
ejpam-6015	226	9	,	,	PUNCT
ejpam-6015	226	10	one	one	PRON
ejpam-6015	226	11	has	have	VERB
ejpam-6015	226	12	ς1	ς1	PROPN
ejpam-6015	226	13	is	be	AUX
ejpam-6015	226	14	a	a	DET
ejpam-6015	226	15	fp	fp	NOUN
ejpam-6015	226	16	of	of	ADP
ejpam-6015	226	17	t	t	PROPN
ejpam-6015	226	18	.	.	PUNCT
ejpam-6015	227	1	otherwise	otherwise	ADV
ejpam-6015	227	2	,	,	PUNCT
ejpam-6015	227	3	suppose	suppose	VERB
ejpam-6015	227	4	ς1	ς1	NOUN
ejpam-6015	227	5	̸=	̸=	PROPN
ejpam-6015	227	6	ς0	ς0	PROPN
ejpam-6015	227	7	and	and	CCONJ
ejpam-6015	227	8	ς1	ς1	PROPN
ejpam-6015	227	9	/∈	/∈	PUNCT
ejpam-6015	228	1	tς1	tς1	INTJ
ejpam-6015	228	2	.	.	PUNCT
ejpam-6015	229	1	so	so	ADV
ejpam-6015	229	2	,	,	PUNCT
ejpam-6015	229	3	dθ(ς0	dθ(ς0	NOUN
ejpam-6015	229	4	,	,	PUNCT
ejpam-6015	229	5	ς1	ς1	NOUN
ejpam-6015	229	6	)	)	PUNCT
ejpam-6015	229	7	>	>	X
ejpam-6015	229	8	0	0	PUNCT
ejpam-6015	230	1	and	and	CCONJ
ejpam-6015	230	2	dθ(ς1	dθ(ς1	ADV
ejpam-6015	230	3	,	,	PUNCT
ejpam-6015	230	4	t	t	NOUN
ejpam-6015	230	5	ς1	ς1	NOUN
ejpam-6015	230	6	)	)	PUNCT
ejpam-6015	230	7	>	>	X
ejpam-6015	231	1	0	0	X
ejpam-6015	231	2	.	.	PUNCT
ejpam-6015	231	3	by	by	ADP
ejpam-6015	231	4	(	(	PUNCT
ejpam-6015	231	5	6	6	NUM
ejpam-6015	231	6	)	)	PUNCT
ejpam-6015	231	7	,	,	PUNCT
ejpam-6015	231	8	we	we	PRON
ejpam-6015	231	9	have	have	VERB
ejpam-6015	231	10	η(hθ(tς0	η(hθ(tς0	NOUN
ejpam-6015	231	11	,	,	PUNCT
ejpam-6015	231	12	t	t	NOUN
ejpam-6015	231	13	ς1	ς1	NOUN
ejpam-6015	231	14	)	)	PUNCT
ejpam-6015	231	15	,	,	PUNCT
ejpam-6015	231	16	dθ(ς0	dθ(ς0	NOUN
ejpam-6015	231	17	,	,	PUNCT
ejpam-6015	231	18	ς1	ς1	NOUN
ejpam-6015	231	19	)	)	PUNCT
ejpam-6015	231	20	)	)	PUNCT
ejpam-6015	231	21	≥	≥	NOUN
ejpam-6015	231	22	0	0	NUM
ejpam-6015	231	23	.	.	PUNCT
ejpam-6015	232	1	(	(	PUNCT
ejpam-6015	232	2	7	7	X
ejpam-6015	232	3	)	)	PUNCT
ejpam-6015	232	4	define	define	VERB
ejpam-6015	232	5	the	the	DET
ejpam-6015	232	6	function	function	NOUN
ejpam-6015	232	7	λ	λ	NOUN
ejpam-6015	232	8	:	:	PUNCT
ejpam-6015	232	9	r×	r×	NOUN
ejpam-6015	232	10	r	r	NOUN
ejpam-6015	232	11	→	→	SYM
ejpam-6015	232	12	r	r	AUX
ejpam-6015	232	13	by	by	ADP
ejpam-6015	232	14	λ(t	λ(t	NOUN
ejpam-6015	232	15	,	,	PUNCT
ejpam-6015	232	16	s	s	PART
ejpam-6015	232	17	)	)	PUNCT
ejpam-6015	232	18	=	=	SYM
ejpam-6015	232	19	{	{	PUNCT
ejpam-6015	232	20	t+η(t	t+η(t	PROPN
ejpam-6015	232	21	,	,	PUNCT
ejpam-6015	232	22	s	s	PART
ejpam-6015	232	23	)	)	PUNCT
ejpam-6015	232	24	s	s	VERB
ejpam-6015	232	25	if	if	SCONJ
ejpam-6015	232	26	t	t	PROPN
ejpam-6015	232	27	,	,	PUNCT
ejpam-6015	232	28	s	s	PART
ejpam-6015	232	29	>	>	X
ejpam-6015	232	30	0	0	NUM
ejpam-6015	232	31	,	,	PUNCT
ejpam-6015	232	32	0	0	NUM
ejpam-6015	232	33	otherwise	otherwise	ADV
ejpam-6015	232	34	.	.	PUNCT
ejpam-6015	233	1	by	by	ADP
ejpam-6015	233	2	(	(	PUNCT
ejpam-6015	233	3	η1	η1	NOUN
ejpam-6015	233	4	)	)	PUNCT
ejpam-6015	233	5	,	,	PUNCT
ejpam-6015	233	6	we	we	PRON
ejpam-6015	233	7	have	have	VERB
ejpam-6015	233	8	0	0	NUM
ejpam-6015	233	9	<	<	X
ejpam-6015	233	10	λ(t	λ(t	PROPN
ejpam-6015	233	11	,	,	PUNCT
ejpam-6015	233	12	s	s	PART
ejpam-6015	233	13	)	)	PUNCT
ejpam-6015	233	14	<	<	X
ejpam-6015	233	15	1	1	NUM
ejpam-6015	233	16	for	for	ADP
ejpam-6015	233	17	every	every	DET
ejpam-6015	233	18	t	t	PROPN
ejpam-6015	233	19	,	,	PUNCT
ejpam-6015	233	20	s	s	PART
ejpam-6015	233	21	>	>	X
ejpam-6015	233	22	0	0	NUM
ejpam-6015	233	23	.	.	PUNCT
ejpam-6015	234	1	(	(	PUNCT
ejpam-6015	234	2	8)	8)	NUM
ejpam-6015	234	3	also	also	ADV
ejpam-6015	234	4	,	,	PUNCT
ejpam-6015	234	5	if	if	SCONJ
ejpam-6015	234	6	η(t	η(t	NOUN
ejpam-6015	234	7	,	,	PUNCT
ejpam-6015	234	8	s	s	PART
ejpam-6015	234	9	)	)	PUNCT
ejpam-6015	234	10	≥	≥	NOUN
ejpam-6015	234	11	0	0	NUM
ejpam-6015	234	12	,	,	PUNCT
ejpam-6015	234	13	then	then	ADV
ejpam-6015	234	14	0	0	NUM
ejpam-6015	234	15	<	<	X
ejpam-6015	234	16	t	t	X
ejpam-6015	234	17	≤	≤	NOUN
ejpam-6015	234	18	sλ(t	sλ(t	NOUN
ejpam-6015	234	19	,	,	PUNCT
ejpam-6015	234	20	s	s	X
ejpam-6015	234	21	)	)	PUNCT
ejpam-6015	234	22	for	for	ADP
ejpam-6015	234	23	all	all	DET
ejpam-6015	234	24	t	t	PROPN
ejpam-6015	234	25	,	,	PUNCT
ejpam-6015	234	26	s	s	PART
ejpam-6015	234	27	>	>	X
ejpam-6015	234	28	0	0	NUM
ejpam-6015	234	29	.	.	PUNCT
ejpam-6015	235	1	(	(	PUNCT
ejpam-6015	235	2	9	9	NUM
ejpam-6015	235	3	)	)	PUNCT
ejpam-6015	235	4	from	from	ADP
ejpam-6015	235	5	(	(	PUNCT
ejpam-6015	235	6	7	7	NUM
ejpam-6015	235	7	)	)	PUNCT
ejpam-6015	235	8	and	and	CCONJ
ejpam-6015	235	9	(	(	PUNCT
ejpam-6015	235	10	8)	8)	NUM
ejpam-6015	235	11	,	,	PUNCT
ejpam-6015	235	12	we	we	PRON
ejpam-6015	235	13	get	get	VERB
ejpam-6015	235	14	0	0	NUM
ejpam-6015	235	15	<	<	X
ejpam-6015	235	16	λ(hθ(tς0	λ(hθ(tς0	PROPN
ejpam-6015	235	17	,	,	PUNCT
ejpam-6015	235	18	t	t	NOUN
ejpam-6015	235	19	ς1	ς1	NOUN
ejpam-6015	235	20	)	)	PUNCT
ejpam-6015	235	21	,	,	PUNCT
ejpam-6015	235	22	dθ(ς0	dθ(ς0	NOUN
ejpam-6015	235	23	,	,	PUNCT
ejpam-6015	235	24	ς1	ς1	NOUN
ejpam-6015	235	25	)	)	PUNCT
ejpam-6015	235	26	)	)	PUNCT
ejpam-6015	235	27	<	<	X
ejpam-6015	236	1	1	1	X
ejpam-6015	236	2	.	.	PUNCT
ejpam-6015	236	3	(	(	PUNCT
ejpam-6015	236	4	10	10	NUM
ejpam-6015	236	5	)	)	PUNCT
ejpam-6015	236	6	since	since	SCONJ
ejpam-6015	236	7	dθ(ς1	dθ(ς1	ADV
ejpam-6015	236	8	,	,	PUNCT
ejpam-6015	236	9	t	t	NOUN
ejpam-6015	236	10	ς1	ς1	NOUN
ejpam-6015	236	11	)	)	PUNCT
ejpam-6015	236	12	>	>	X
ejpam-6015	236	13	0	0	NUM
ejpam-6015	236	14	,	,	PUNCT
ejpam-6015	236	15	by	by	ADP
ejpam-6015	236	16	using	use	VERB
ejpam-6015	236	17	(	(	PUNCT
ejpam-6015	236	18	10	10	NUM
ejpam-6015	236	19	)	)	PUNCT
ejpam-6015	236	20	,	,	PUNCT
ejpam-6015	236	21	we	we	PRON
ejpam-6015	236	22	have	have	VERB
ejpam-6015	236	23	dθ(ς1	dθ(ς1	ADV
ejpam-6015	236	24	,	,	PUNCT
ejpam-6015	236	25	t	t	PROPN
ejpam-6015	236	26	ς1	ς1	NOUN
ejpam-6015	236	27	)	)	PUNCT
ejpam-6015	236	28	<	<	X
ejpam-6015	236	29	1√	1√	PROPN
ejpam-6015	236	30	λ(hθ(tς0	λ(hθ(tς0	PROPN
ejpam-6015	236	31	,	,	PUNCT
ejpam-6015	236	32	t	t	NOUN
ejpam-6015	236	33	ς1	ς1	NOUN
ejpam-6015	236	34	)	)	PUNCT
ejpam-6015	236	35	,	,	PUNCT
ejpam-6015	236	36	dθ(ς0	dθ(ς0	NOUN
ejpam-6015	236	37	,	,	PUNCT
ejpam-6015	236	38	ς1	ς1	NOUN
ejpam-6015	236	39	)	)	PUNCT
ejpam-6015	236	40	)	)	PUNCT
ejpam-6015	237	1	dθ(ς1	dθ(ς1	ADV
ejpam-6015	237	2	,	,	PUNCT
ejpam-6015	237	3	t	t	NOUN
ejpam-6015	237	4	ς1	ς1	NOUN
ejpam-6015	237	5	)	)	PUNCT
ejpam-6015	237	6	.	.	PUNCT
ejpam-6015	238	1	h.	h.	PROPN
ejpam-6015	238	2	aydi	aydi	VERB
ejpam-6015	238	3	et	et	PROPN
ejpam-6015	238	4	al	al	PROPN
ejpam-6015	238	5	.	.	PUNCT
ejpam-6015	238	6	/	/	SYM
ejpam-6015	238	7	eur	eur	PROPN
ejpam-6015	238	8	.	.	PUNCT
ejpam-6015	239	1	j.	j.	PROPN
ejpam-6015	239	2	pure	pure	PROPN
ejpam-6015	239	3	appl	appl	PROPN
ejpam-6015	239	4	.	.	PROPN
ejpam-6015	239	5	math	math	PROPN
ejpam-6015	239	6	,	,	PUNCT
ejpam-6015	239	7	18	18	NUM
ejpam-6015	239	8	(	(	PUNCT
ejpam-6015	239	9	2	2	NUM
ejpam-6015	239	10	)	)	PUNCT
ejpam-6015	239	11	(	(	PUNCT
ejpam-6015	239	12	2025	2025	NUM
ejpam-6015	239	13	)	)	PUNCT
ejpam-6015	239	14	,	,	PUNCT
ejpam-6015	239	15	6015	6015	NUM
ejpam-6015	239	16	10	10	NUM
ejpam-6015	239	17	of	of	ADP
ejpam-6015	239	18	14	14	NUM
ejpam-6015	239	19	using	use	VERB
ejpam-6015	239	20	lemma	lemma	PROPN
ejpam-6015	239	21	2	2	NUM
ejpam-6015	239	22	,	,	PUNCT
ejpam-6015	239	23	there	there	PRON
ejpam-6015	239	24	is	be	VERB
ejpam-6015	239	25	ς2	ς2	PROPN
ejpam-6015	239	26	∈	∈	PROPN
ejpam-6015	239	27	tς1	tς1	PROPN
ejpam-6015	239	28	so	so	SCONJ
ejpam-6015	239	29	that	that	SCONJ
ejpam-6015	239	30	dθ(ς1	dθ(ς1	ADV
ejpam-6015	239	31	,	,	PUNCT
ejpam-6015	239	32	ς2	ς2	PROPN
ejpam-6015	239	33	)	)	PUNCT
ejpam-6015	239	34	<	<	X
ejpam-6015	239	35	1√	1√	PROPN
ejpam-6015	239	36	λ(hθ(tς0	λ(hθ(tς0	PROPN
ejpam-6015	239	37	,	,	PUNCT
ejpam-6015	239	38	t	t	NOUN
ejpam-6015	239	39	ς1	ς1	NOUN
ejpam-6015	239	40	)	)	PUNCT
ejpam-6015	239	41	,	,	PUNCT
ejpam-6015	239	42	dθ(ς0	dθ(ς0	NOUN
ejpam-6015	239	43	,	,	PUNCT
ejpam-6015	239	44	ς1	ς1	NOUN
ejpam-6015	239	45	)	)	PUNCT
ejpam-6015	239	46	)	)	PUNCT
ejpam-6015	240	1	dθ(ς1	dθ(ς1	ADV
ejpam-6015	240	2	,	,	PUNCT
ejpam-6015	240	3	t	t	NOUN
ejpam-6015	240	4	ς1	ς1	NOUN
ejpam-6015	240	5	)	)	PUNCT
ejpam-6015	240	6	.	.	PUNCT
ejpam-6015	241	1	(	(	PUNCT
ejpam-6015	241	2	11	11	X
ejpam-6015	241	3	)	)	PUNCT
ejpam-6015	241	4	it	it	PRON
ejpam-6015	241	5	follows	follow	VERB
ejpam-6015	241	6	that	that	SCONJ
ejpam-6015	241	7	dθ(ς1	dθ(ς1	ADV
ejpam-6015	241	8	,	,	PUNCT
ejpam-6015	241	9	t	t	NOUN
ejpam-6015	241	10	ς1	ς1	NOUN
ejpam-6015	241	11	)	)	PUNCT
ejpam-6015	241	12	≤	≤	NOUN
ejpam-6015	241	13	dθ(ς0	dθ(ς0	NOUN
ejpam-6015	241	14	,	,	PUNCT
ejpam-6015	241	15	ς1)λ(hθ(tς0	ς1)λ(hθ(tς0	AUX
ejpam-6015	241	16	,	,	PUNCT
ejpam-6015	241	17	t	t	NOUN
ejpam-6015	241	18	ς1	ς1	NOUN
ejpam-6015	241	19	)	)	PUNCT
ejpam-6015	241	20	,	,	PUNCT
ejpam-6015	241	21	dθ(ς0	dθ(ς0	NOUN
ejpam-6015	241	22	,	,	PUNCT
ejpam-6015	241	23	ς1	ς1	NOUN
ejpam-6015	241	24	)	)	PUNCT
ejpam-6015	241	25	)	)	PUNCT
ejpam-6015	241	26	.	.	PUNCT
ejpam-6015	242	1	(	(	PUNCT
ejpam-6015	242	2	12	12	X
ejpam-6015	242	3	)	)	PUNCT
ejpam-6015	242	4	combining	combine	VERB
ejpam-6015	242	5	(	(	PUNCT
ejpam-6015	242	6	11	11	NUM
ejpam-6015	242	7	)	)	PUNCT
ejpam-6015	242	8	and	and	CCONJ
ejpam-6015	242	9	(	(	PUNCT
ejpam-6015	242	10	12	12	NUM
ejpam-6015	242	11	)	)	PUNCT
ejpam-6015	242	12	,	,	PUNCT
ejpam-6015	242	13	we	we	PRON
ejpam-6015	242	14	get	get	VERB
ejpam-6015	242	15	dθ(ς1	dθ(ς1	ADV
ejpam-6015	242	16	,	,	PUNCT
ejpam-6015	242	17	ς2	ς2	PROPN
ejpam-6015	242	18	)	)	PUNCT
ejpam-6015	242	19	≤	≤	NOUN
ejpam-6015	242	20	√	√	NUM
ejpam-6015	242	21	λ(hθ(tς0	λ(hθ(tς0	PROPN
ejpam-6015	242	22	,	,	PUNCT
ejpam-6015	242	23	t	t	NOUN
ejpam-6015	242	24	ς1	ς1	NOUN
ejpam-6015	242	25	)	)	PUNCT
ejpam-6015	242	26	,	,	PUNCT
ejpam-6015	242	27	dθ(ς0	dθ(ς0	NOUN
ejpam-6015	242	28	,	,	PUNCT
ejpam-6015	242	29	ς1))dθ(ς0	ς1))dθ(ς0	NUM
ejpam-6015	242	30	,	,	PUNCT
ejpam-6015	242	31	ς1	ς1	NOUN
ejpam-6015	242	32	)	)	PUNCT
ejpam-6015	242	33	.	.	PUNCT
ejpam-6015	243	1	note	note	VERB
ejpam-6015	243	2	that	that	SCONJ
ejpam-6015	243	3	ς2	ς2	PROPN
ejpam-6015	243	4	̸=	̸=	PROPN
ejpam-6015	243	5	ς1	ς1	NOUN
ejpam-6015	243	6	because	because	SCONJ
ejpam-6015	243	7	ς1	ς1	NOUN
ejpam-6015	243	8	/∈	/∈	PUNCT
ejpam-6015	244	1	tς1	tς1	INTJ
ejpam-6015	244	2	.	.	PUNCT
ejpam-6015	245	1	when	when	SCONJ
ejpam-6015	245	2	ς2	ς2	PROPN
ejpam-6015	245	3	∈	∈	PROPN
ejpam-6015	245	4	tς2	tς2	NOUN
ejpam-6015	245	5	,	,	PUNCT
ejpam-6015	245	6	ς2	ς2	PROPN
ejpam-6015	245	7	is	be	AUX
ejpam-6015	245	8	a	a	DET
ejpam-6015	245	9	fp	fp	NOUN
ejpam-6015	245	10	of	of	ADP
ejpam-6015	245	11	t	t	PROPN
ejpam-6015	245	12	.	.	PUNCT
ejpam-6015	246	1	suppose	suppose	VERB
ejpam-6015	246	2	ς2	ς2	PROPN
ejpam-6015	246	3	/∈	/∈	PUNCT
ejpam-6015	246	4	tς2	tς2	NOUN
ejpam-6015	246	5	.	.	PUNCT
ejpam-6015	247	1	hence	hence	ADV
ejpam-6015	247	2	,	,	PUNCT
ejpam-6015	247	3	by	by	ADP
ejpam-6015	247	4	(	(	PUNCT
ejpam-6015	247	5	6	6	NUM
ejpam-6015	247	6	)	)	PUNCT
ejpam-6015	247	7	,	,	PUNCT
ejpam-6015	247	8	η(hθ(tς1	η(hθ(tς1	PROPN
ejpam-6015	247	9	,	,	PUNCT
ejpam-6015	247	10	t	t	PROPN
ejpam-6015	247	11	ς2	ς2	PROPN
ejpam-6015	247	12	)	)	PUNCT
ejpam-6015	247	13	,	,	PUNCT
ejpam-6015	247	14	dθ(ς1	dθ(ς1	ADV
ejpam-6015	247	15	,	,	PUNCT
ejpam-6015	247	16	ς2	ς2	PROPN
ejpam-6015	247	17	)	)	PUNCT
ejpam-6015	247	18	)	)	PUNCT
ejpam-6015	247	19	≥	≥	NOUN
ejpam-6015	247	20	0	0	NUM
ejpam-6015	247	21	.	.	PUNCT
ejpam-6015	248	1	since	since	SCONJ
ejpam-6015	248	2	dθ(ς2	dθ(ς2	PROPN
ejpam-6015	248	3	,	,	PUNCT
ejpam-6015	248	4	t	t	PROPN
ejpam-6015	248	5	ς2	ς2	PROPN
ejpam-6015	248	6	)	)	PUNCT
ejpam-6015	248	7	>	>	X
ejpam-6015	248	8	0	0	NUM
ejpam-6015	248	9	,	,	PUNCT
ejpam-6015	248	10	by	by	ADP
ejpam-6015	248	11	using	use	VERB
ejpam-6015	248	12	(	(	PUNCT
ejpam-6015	248	13	10	10	NUM
ejpam-6015	248	14	)	)	PUNCT
ejpam-6015	248	15	,	,	PUNCT
ejpam-6015	248	16	we	we	PRON
ejpam-6015	248	17	have	have	VERB
ejpam-6015	248	18	dθ(ς2	dθ(ς2	NOUN
ejpam-6015	248	19	,	,	PUNCT
ejpam-6015	248	20	t	t	PROPN
ejpam-6015	248	21	ς2	ς2	PROPN
ejpam-6015	248	22	)	)	PUNCT
ejpam-6015	248	23	<	<	X
ejpam-6015	248	24	1√	1√	PROPN
ejpam-6015	248	25	λ(hθ(tς1	λ(hθ(tς1	PROPN
ejpam-6015	248	26	,	,	PUNCT
ejpam-6015	248	27	t	t	PROPN
ejpam-6015	248	28	ς2	ς2	PROPN
ejpam-6015	248	29	)	)	PUNCT
ejpam-6015	248	30	,	,	PUNCT
ejpam-6015	248	31	dθ(ς1	dθ(ς1	ADV
ejpam-6015	248	32	,	,	PUNCT
ejpam-6015	248	33	ς2	ς2	PROPN
ejpam-6015	248	34	)	)	PUNCT
ejpam-6015	248	35	)	)	PUNCT
ejpam-6015	249	1	dθ(ς1	dθ(ς1	ADV
ejpam-6015	249	2	,	,	PUNCT
ejpam-6015	249	3	t	t	PROPN
ejpam-6015	249	4	ς2	ς2	PROPN
ejpam-6015	249	5	)	)	PUNCT
ejpam-6015	249	6	.	.	PUNCT
ejpam-6015	250	1	lemma	lemma	PROPN
ejpam-6015	250	2	2	2	PROPN
ejpam-6015	250	3	implies	imply	VERB
ejpam-6015	250	4	the	the	DET
ejpam-6015	250	5	existence	existence	NOUN
ejpam-6015	250	6	of	of	ADP
ejpam-6015	250	7	a	a	DET
ejpam-6015	250	8	point	point	NOUN
ejpam-6015	250	9	ς3	ς3	NOUN
ejpam-6015	250	10	∈	∈	PROPN
ejpam-6015	250	11	tς2	tς2	NOUN
ejpam-6015	250	12	such	such	ADJ
ejpam-6015	250	13	that	that	DET
ejpam-6015	250	14	dθ(ς2	dθ(ς2	NOUN
ejpam-6015	250	15	,	,	PUNCT
ejpam-6015	250	16	ς3	ς3	NOUN
ejpam-6015	250	17	)	)	PUNCT
ejpam-6015	250	18	<	<	X
ejpam-6015	250	19	1√	1√	PROPN
ejpam-6015	250	20	λ(hθ(tς1	λ(hθ(tς1	PROPN
ejpam-6015	250	21	,	,	PUNCT
ejpam-6015	250	22	t	t	PROPN
ejpam-6015	250	23	ς2	ς2	PROPN
ejpam-6015	250	24	)	)	PUNCT
ejpam-6015	250	25	,	,	PUNCT
ejpam-6015	250	26	dθ(ς1	dθ(ς1	ADV
ejpam-6015	250	27	,	,	PUNCT
ejpam-6015	250	28	ς2	ς2	PROPN
ejpam-6015	250	29	)	)	PUNCT
ejpam-6015	250	30	)	)	PUNCT
ejpam-6015	250	31	dθ(ς2	dθ(ς2	PROPN
ejpam-6015	250	32	,	,	PUNCT
ejpam-6015	250	33	t	t	PROPN
ejpam-6015	250	34	ς2	ς2	PROPN
ejpam-6015	250	35	)	)	PUNCT
ejpam-6015	250	36	.	.	PUNCT
ejpam-6015	251	1	similarly	similarly	ADV
ejpam-6015	251	2	,	,	PUNCT
ejpam-6015	251	3	we	we	PRON
ejpam-6015	251	4	get	get	VERB
ejpam-6015	251	5	dθ(ς2	dθ(ς2	NOUN
ejpam-6015	251	6	,	,	PUNCT
ejpam-6015	251	7	ς3	ς3	NOUN
ejpam-6015	251	8	)	)	PUNCT
ejpam-6015	251	9	≤	≤	NOUN
ejpam-6015	251	10	√	√	NUM
ejpam-6015	251	11	λ(hθ(tς1	λ(hθ(tς1	PROPN
ejpam-6015	251	12	,	,	PUNCT
ejpam-6015	251	13	t	t	PROPN
ejpam-6015	251	14	ς2	ς2	PROPN
ejpam-6015	251	15	)	)	PUNCT
ejpam-6015	251	16	,	,	PUNCT
ejpam-6015	251	17	dθ(ς1	dθ(ς1	ADV
ejpam-6015	251	18	,	,	PUNCT
ejpam-6015	251	19	ς2))dθ(ς1	ς2))dθ(ς1	NUM
ejpam-6015	251	20	,	,	PUNCT
ejpam-6015	251	21	ς2	ς2	PROPN
ejpam-6015	251	22	)	)	PUNCT
ejpam-6015	251	23	.	.	PUNCT
ejpam-6015	252	1	continuing	continue	VERB
ejpam-6015	252	2	the	the	DET
ejpam-6015	252	3	same	same	ADJ
ejpam-6015	252	4	work	work	NOUN
ejpam-6015	252	5	,	,	PUNCT
ejpam-6015	252	6	we	we	PRON
ejpam-6015	252	7	build	build	VERB
ejpam-6015	252	8	{	{	PUNCT
ejpam-6015	252	9	ςn	ςn	NOUN
ejpam-6015	252	10	}	}	PUNCT
ejpam-6015	252	11	in	in	ADP
ejpam-6015	252	12	x	x	PROPN
ejpam-6015	252	13	so	so	SCONJ
ejpam-6015	252	14	that	that	SCONJ
ejpam-6015	252	15	for	for	ADP
ejpam-6015	252	16	any	any	DET
ejpam-6015	252	17	n	n	PRON
ejpam-6015	252	18	≥	≥	NOUN
ejpam-6015	252	19	1	1	NUM
ejpam-6015	252	20	,	,	PUNCT
ejpam-6015	252	21	dθ(ςn	dθ(ςn	NOUN
ejpam-6015	252	22	,	,	PUNCT
ejpam-6015	252	23	ςn+1	ςn+1	NUM
ejpam-6015	252	24	)	)	PUNCT
ejpam-6015	252	25	≤	≤	NOUN
ejpam-6015	252	26	√	√	PUNCT
ejpam-6015	252	27	λ(hθ(tςn−1	λ(hθ(tςn−1	PROPN
ejpam-6015	252	28	,	,	PUNCT
ejpam-6015	252	29	t	t	PROPN
ejpam-6015	252	30	ςn	ςn	PROPN
ejpam-6015	252	31	)	)	PUNCT
ejpam-6015	252	32	,	,	PUNCT
ejpam-6015	252	33	dθ(ςn−1	dθ(ςn−1	PROPN
ejpam-6015	252	34	,	,	PUNCT
ejpam-6015	252	35	ςn))dθ(ςn−1	ςn))dθ(ςn−1	NUM
ejpam-6015	252	36	,	,	PUNCT
ejpam-6015	252	37	ςn	ςn	NOUN
ejpam-6015	252	38	)	)	PUNCT
ejpam-6015	252	39	.	.	PUNCT
ejpam-6015	253	1	(	(	PUNCT
ejpam-6015	253	2	13	13	NUM
ejpam-6015	253	3	)	)	PUNCT
ejpam-6015	253	4	from	from	ADP
ejpam-6015	253	5	(	(	PUNCT
ejpam-6015	253	6	8)	8)	NUM
ejpam-6015	253	7	and	and	CCONJ
ejpam-6015	253	8	(	(	PUNCT
ejpam-6015	253	9	13	13	NUM
ejpam-6015	253	10	)	)	PUNCT
ejpam-6015	253	11	,	,	PUNCT
ejpam-6015	253	12	we	we	PRON
ejpam-6015	253	13	get	get	VERB
ejpam-6015	253	14	0	0	NUM
ejpam-6015	253	15	<	<	X
ejpam-6015	253	16	dθ(ςn	dθ(ςn	NOUN
ejpam-6015	253	17	,	,	PUNCT
ejpam-6015	253	18	ςn+1	ςn+1	NUM
ejpam-6015	253	19	)	)	PUNCT
ejpam-6015	253	20	<	<	X
ejpam-6015	253	21	dθ(ςn−1	dθ(ςn−1	PROPN
ejpam-6015	253	22	,	,	PUNCT
ejpam-6015	253	23	ςn	ςn	NOUN
ejpam-6015	253	24	)	)	PUNCT
ejpam-6015	253	25	for	for	ADP
ejpam-6015	253	26	all	all	DET
ejpam-6015	253	27	n	n	CCONJ
ejpam-6015	253	28	,	,	PUNCT
ejpam-6015	253	29	which	which	PRON
ejpam-6015	253	30	yields	yield	VERB
ejpam-6015	253	31	that	that	SCONJ
ejpam-6015	253	32	{	{	PUNCT
ejpam-6015	253	33	dθ(ςn−1	dθ(ςn−1	PROPN
ejpam-6015	253	34	,	,	PUNCT
ejpam-6015	253	35	ςn	ςn	NOUN
ejpam-6015	253	36	)	)	PUNCT
ejpam-6015	253	37	}	}	PUNCT
ejpam-6015	253	38	is	be	AUX
ejpam-6015	253	39	non	non	ADJ
ejpam-6015	253	40	-	-	ADJ
ejpam-6015	253	41	increasing	increase	VERB
ejpam-6015	253	42	and	and	CCONJ
ejpam-6015	253	43	positive	positive	ADJ
ejpam-6015	253	44	,	,	PUNCT
ejpam-6015	253	45	so	so	CCONJ
ejpam-6015	253	46	it	it	PRON
ejpam-6015	253	47	is	be	AUX
ejpam-6015	253	48	convergent	convergent	ADJ
ejpam-6015	253	49	.	.	PUNCT
ejpam-6015	254	1	also	also	ADV
ejpam-6015	254	2	,	,	PUNCT
ejpam-6015	254	3	0	0	PUNCT
ejpam-6015	254	4	<	<	X
ejpam-6015	254	5	hθ(tςn−1	hθ(tςn−1	PROPN
ejpam-6015	254	6	,	,	PUNCT
ejpam-6015	254	7	t	t	NOUN
ejpam-6015	254	8	ςn	ςn	NOUN
ejpam-6015	254	9	)	)	PUNCT
ejpam-6015	254	10	<	<	X
ejpam-6015	254	11	dθ(ςn−1	dθ(ςn−1	PROPN
ejpam-6015	254	12	,	,	PUNCT
ejpam-6015	254	13	ςn	ςn	NOUN
ejpam-6015	254	14	)	)	PUNCT
ejpam-6015	254	15	,	,	PUNCT
ejpam-6015	254	16	for	for	ADP
ejpam-6015	254	17	all	all	DET
ejpam-6015	254	18	n	n	CCONJ
ejpam-6015	254	19	,	,	PUNCT
ejpam-6015	254	20	which	which	PRON
ejpam-6015	254	21	yields	yield	VERB
ejpam-6015	254	22	that	that	SCONJ
ejpam-6015	254	23	{	{	PUNCT
ejpam-6015	254	24	hθ(tςn−1	hθ(tςn−1	PROPN
ejpam-6015	254	25	,	,	PUNCT
ejpam-6015	254	26	t	t	PROPN
ejpam-6015	254	27	ςn	ςn	NOUN
ejpam-6015	254	28	)	)	PUNCT
ejpam-6015	254	29	}	}	PUNCT
ejpam-6015	254	30	is	be	AUX
ejpam-6015	254	31	bounded	bound	VERB
ejpam-6015	254	32	.	.	PUNCT
ejpam-6015	255	1	from	from	ADP
ejpam-6015	255	2	(	(	PUNCT
ejpam-6015	255	3	η2	η2	PROPN
ejpam-6015	255	4	)	)	PUNCT
ejpam-6015	255	5	,	,	PUNCT
ejpam-6015	255	6	lim	lim	PROPN
ejpam-6015	255	7	sup	sup	PROPN
ejpam-6015	255	8	n→+∞	n→+∞	PROPN
ejpam-6015	255	9	λ(hθ(tςn−1	λ(hθ(tςn−1	PROPN
ejpam-6015	255	10	,	,	PUNCT
ejpam-6015	255	11	t	t	PROPN
ejpam-6015	255	12	ςn	ςn	PROPN
ejpam-6015	255	13	)	)	PUNCT
ejpam-6015	255	14	,	,	PUNCT
ejpam-6015	255	15	dθ(ςn−1	dθ(ςn−1	PROPN
ejpam-6015	255	16	,	,	PUNCT
ejpam-6015	255	17	ςn	ςn	NOUN
ejpam-6015	255	18	)	)	PUNCT
ejpam-6015	255	19	)	)	PUNCT
ejpam-6015	256	1	<	<	X
ejpam-6015	257	1	1	1	X
ejpam-6015	257	2	.	.	PUNCT
ejpam-6015	257	3	(	(	PUNCT
ejpam-6015	257	4	14	14	NUM
ejpam-6015	257	5	)	)	PUNCT
ejpam-6015	257	6	let	let	VERB
ejpam-6015	257	7	λn	λn	NOUN
ejpam-6015	257	8	=	=	PUNCT
ejpam-6015	257	9	√	√	PROPN
ejpam-6015	257	10	λ(hθ(tςn−1	λ(hθ(tςn−1	PROPN
ejpam-6015	257	11	,	,	PUNCT
ejpam-6015	257	12	t	t	PROPN
ejpam-6015	257	13	ςn	ςn	PROPN
ejpam-6015	257	14	)	)	PUNCT
ejpam-6015	257	15	,	,	PUNCT
ejpam-6015	257	16	dθ(ςn−1	dθ(ςn−1	PROPN
ejpam-6015	257	17	,	,	PUNCT
ejpam-6015	257	18	ςn	ςn	NOUN
ejpam-6015	257	19	)	)	PUNCT
ejpam-6015	257	20	)	)	PUNCT
ejpam-6015	257	21	,	,	PUNCT
ejpam-6015	257	22	∀n	∀n	NUM
ejpam-6015	257	23	≥	≥	NOUN
ejpam-6015	257	24	1	1	NUM
ejpam-6015	257	25	.	.	PUNCT
ejpam-6015	257	26	(	(	PUNCT
ejpam-6015	257	27	15	15	NUM
ejpam-6015	257	28	)	)	PUNCT
ejpam-6015	257	29	from	from	ADP
ejpam-6015	257	30	(	(	PUNCT
ejpam-6015	257	31	13	13	NUM
ejpam-6015	257	32	)	)	PUNCT
ejpam-6015	257	33	,	,	PUNCT
ejpam-6015	257	34	we	we	PRON
ejpam-6015	257	35	get	get	VERB
ejpam-6015	257	36	dθ(ςn	dθ(ςn	NOUN
ejpam-6015	257	37	,	,	PUNCT
ejpam-6015	257	38	ςn+1	ςn+1	NUM
ejpam-6015	257	39	)	)	PUNCT
ejpam-6015	257	40	≤	≤	NOUN
ejpam-6015	257	41	λndθ(ςn−1	λndθ(ςn−1	NOUN
ejpam-6015	257	42	,	,	PUNCT
ejpam-6015	257	43	ςn	ςn	NOUN
ejpam-6015	257	44	)	)	PUNCT
ejpam-6015	257	45	,	,	PUNCT
ejpam-6015	257	46	∀n	∀n	NUM
ejpam-6015	257	47	≥	≥	NOUN
ejpam-6015	257	48	1	1	NUM
ejpam-6015	257	49	.	.	PUNCT
ejpam-6015	258	1	(	(	PUNCT
ejpam-6015	258	2	16	16	NUM
ejpam-6015	258	3	)	)	PUNCT
ejpam-6015	258	4	h.	h.	PROPN
ejpam-6015	258	5	aydi	aydi	VERB
ejpam-6015	258	6	et	et	PROPN
ejpam-6015	258	7	al	al	PROPN
ejpam-6015	258	8	.	.	PUNCT
ejpam-6015	258	9	/	/	SYM
ejpam-6015	258	10	eur	eur	PROPN
ejpam-6015	258	11	.	.	PUNCT
ejpam-6015	259	1	j.	j.	PROPN
ejpam-6015	259	2	pure	pure	PROPN
ejpam-6015	259	3	appl	appl	PROPN
ejpam-6015	259	4	.	.	PROPN
ejpam-6015	259	5	math	math	PROPN
ejpam-6015	259	6	,	,	PUNCT
ejpam-6015	259	7	18	18	NUM
ejpam-6015	259	8	(	(	PUNCT
ejpam-6015	259	9	2	2	NUM
ejpam-6015	259	10	)	)	PUNCT
ejpam-6015	259	11	(	(	PUNCT
ejpam-6015	259	12	2025	2025	NUM
ejpam-6015	259	13	)	)	PUNCT
ejpam-6015	259	14	,	,	PUNCT
ejpam-6015	259	15	6015	6015	NUM
ejpam-6015	259	16	11	11	NUM
ejpam-6015	259	17	of	of	ADP
ejpam-6015	259	18	14	14	NUM
ejpam-6015	259	19	by	by	ADP
ejpam-6015	259	20	(	(	PUNCT
ejpam-6015	259	21	14	14	NUM
ejpam-6015	259	22	)	)	PUNCT
ejpam-6015	259	23	,	,	PUNCT
ejpam-6015	259	24	there	there	PRON
ejpam-6015	259	25	are	be	VERB
ejpam-6015	259	26	α	α	PRON
ejpam-6015	259	27	∈	∈	PROPN
ejpam-6015	259	28	(	(	PUNCT
ejpam-6015	259	29	0	0	NUM
ejpam-6015	259	30	,	,	PUNCT
ejpam-6015	259	31	1	1	NUM
ejpam-6015	259	32	)	)	PUNCT
ejpam-6015	259	33	and	and	CCONJ
ejpam-6015	259	34	n0	n0	NUM
ejpam-6015	259	35	∈	∈	PROPN
ejpam-6015	259	36	n	n	PRON
ejpam-6015	259	37	so	so	ADV
ejpam-6015	259	38	that	that	SCONJ
ejpam-6015	259	39	λn	λn	PROPN
ejpam-6015	259	40	≤	≤	NUM
ejpam-6015	259	41	α	α	NOUN
ejpam-6015	259	42	,	,	PUNCT
ejpam-6015	259	43	∀n	∀n	NUM
ejpam-6015	259	44	≥	≥	NOUN
ejpam-6015	259	45	n0	n0	NUM
ejpam-6015	259	46	.	.	PUNCT
ejpam-6015	260	1	hence	hence	ADV
ejpam-6015	260	2	,	,	PUNCT
ejpam-6015	260	3	by	by	ADP
ejpam-6015	260	4	(	(	PUNCT
ejpam-6015	260	5	16	16	NUM
ejpam-6015	260	6	)	)	PUNCT
ejpam-6015	260	7	,	,	PUNCT
ejpam-6015	260	8	we	we	PRON
ejpam-6015	260	9	get	get	VERB
ejpam-6015	260	10	dθ(ςn	dθ(ςn	NOUN
ejpam-6015	260	11	,	,	PUNCT
ejpam-6015	260	12	ςn+1	ςn+1	NUM
ejpam-6015	260	13	)	)	PUNCT
ejpam-6015	260	14	≤	≤	NOUN
ejpam-6015	260	15	αdθ(ςn−1	αdθ(ςn−1	PROPN
ejpam-6015	260	16	,	,	PUNCT
ejpam-6015	260	17	ςn	ςn	NOUN
ejpam-6015	260	18	)	)	PUNCT
ejpam-6015	260	19	,	,	PUNCT
ejpam-6015	260	20	∀n	∀n	NUM
ejpam-6015	260	21	≥	≥	PROPN
ejpam-6015	260	22	n0	n0	NUM
ejpam-6015	260	23	.	.	PUNCT
ejpam-6015	261	1	thus	thus	ADV
ejpam-6015	261	2	,	,	PUNCT
ejpam-6015	261	3	dθ(ςn	dθ(ςn	NOUN
ejpam-6015	261	4	,	,	PUNCT
ejpam-6015	261	5	ςn+1	ςn+1	NUM
ejpam-6015	261	6	)	)	PUNCT
ejpam-6015	261	7	≤	≤	NUM
ejpam-6015	261	8	αn−n0	αn−n0	NOUN
ejpam-6015	261	9	+	+	PROPN
ejpam-6015	261	10	1dθ(ςn0−1	1dθ(ςn0−1	NUM
ejpam-6015	261	11	,	,	PUNCT
ejpam-6015	261	12	ςn0	ςn0	NOUN
ejpam-6015	261	13	)	)	PUNCT
ejpam-6015	261	14	,	,	PUNCT
ejpam-6015	262	1	∀n	∀n	NUM
ejpam-6015	262	2	≥	≥	PROPN
ejpam-6015	262	3	n0	n0	NUM
ejpam-6015	262	4	.	.	PUNCT
ejpam-6015	263	1	moreover	moreover	ADV
ejpam-6015	263	2	,	,	PUNCT
ejpam-6015	263	3	by	by	ADP
ejpam-6015	263	4	(	(	PUNCT
ejpam-6015	263	5	1	1	NUM
ejpam-6015	263	6	)	)	PUNCT
ejpam-6015	263	7	,	,	PUNCT
ejpam-6015	263	8	and	and	CCONJ
ejpam-6015	263	9	sinh(t	sinh(t	NOUN
ejpam-6015	263	10	)	)	PUNCT
ejpam-6015	263	11	≥	≥	PROPN
ejpam-6015	263	12	t	t	PROPN
ejpam-6015	263	13	for	for	ADP
ejpam-6015	263	14	all	all	DET
ejpam-6015	263	15	t	t	PROPN
ejpam-6015	263	16	≥	≥	NOUN
ejpam-6015	263	17	0	0	NUM
ejpam-6015	263	18	,	,	PUNCT
ejpam-6015	263	19	one	one	PRON
ejpam-6015	263	20	gets	get	VERB
ejpam-6015	263	21	d(ςn	d(ςn	NOUN
ejpam-6015	263	22	,	,	PUNCT
ejpam-6015	263	23	ςn+1	ςn+1	NUM
ejpam-6015	263	24	)	)	PUNCT
ejpam-6015	263	25	≤	≤	NOUN
ejpam-6015	264	1	(	(	PUNCT
ejpam-6015	264	2	α	α	NOUN
ejpam-6015	264	3	1	1	NUM
ejpam-6015	264	4	τ	τ	NOUN
ejpam-6015	264	5	)	)	PUNCT
ejpam-6015	264	6	n−n0	n−n0	NOUN
ejpam-6015	264	7	+	+	SYM
ejpam-6015	264	8	1	1	NUM
ejpam-6015	264	9	(	(	PUNCT
ejpam-6015	264	10	dθ(ςn0−1	dθ(ςn0−1	ADJ
ejpam-6015	264	11	,	,	PUNCT
ejpam-6015	264	12	ςn0	ςn0	NOUN
ejpam-6015	264	13	)	)	PUNCT
ejpam-6015	264	14	c	c	NOUN
ejpam-6015	264	15	)	)	PUNCT
ejpam-6015	264	16	1	1	NUM
ejpam-6015	264	17	τ	τ	PROPN
ejpam-6015	264	18	,	,	PUNCT
ejpam-6015	264	19	∀n	∀n	NUM
ejpam-6015	264	20	≥	≥	NOUN
ejpam-6015	264	21	n0	n0	NUM
ejpam-6015	264	22	.	.	PUNCT
ejpam-6015	265	1	now	now	ADV
ejpam-6015	265	2	,	,	PUNCT
ejpam-6015	265	3	for	for	ADP
ejpam-6015	265	4	m	m	PROPN
ejpam-6015	265	5	>	>	X
ejpam-6015	265	6	n	n	CCONJ
ejpam-6015	265	7	≥	≥	PROPN
ejpam-6015	265	8	n0	n0	NUM
ejpam-6015	265	9	,	,	PUNCT
ejpam-6015	265	10	we	we	PRON
ejpam-6015	265	11	have	have	AUX
ejpam-6015	265	12	d(ςn	d(ςn	VERB
ejpam-6015	265	13	,	,	PUNCT
ejpam-6015	265	14	ςm	ςm	NOUN
ejpam-6015	265	15	)	)	PUNCT
ejpam-6015	265	16	≤	≤	NOUN
ejpam-6015	266	1	m−1∑	m−1∑	NUM
ejpam-6015	266	2	i	i	NOUN
ejpam-6015	266	3	=	=	PROPN
ejpam-6015	266	4	n	n	PRON
ejpam-6015	266	5	d(ςi	d(ςi	PROPN
ejpam-6015	266	6	,	,	PUNCT
ejpam-6015	266	7	ςi+1	ςi+1	NUM
ejpam-6015	266	8	)	)	PUNCT
ejpam-6015	266	9	≤	≤	NOUN
ejpam-6015	266	10	(	(	PUNCT
ejpam-6015	266	11	dθ(ςn0−1	dθ(ςn0−1	ADJ
ejpam-6015	266	12	,	,	PUNCT
ejpam-6015	266	13	ςn0	ςn0	NOUN
ejpam-6015	266	14	)	)	PUNCT
ejpam-6015	266	15	c	c	NOUN
ejpam-6015	266	16	)	)	PUNCT
ejpam-6015	266	17	1	1	NUM
ejpam-6015	266	18	τ	τ	PUNCT
ejpam-6015	266	19	+	+	ADJ
ejpam-6015	266	20	∞∑	∞∑	NUM
ejpam-6015	266	21	i	i	NOUN
ejpam-6015	266	22	=	=	NOUN
ejpam-6015	266	23	n	n	X
ejpam-6015	266	24	(	(	PUNCT
ejpam-6015	266	25	α	α	NOUN
ejpam-6015	266	26	1	1	NUM
ejpam-6015	266	27	τ	τ	NOUN
ejpam-6015	266	28	)	)	PUNCT
ejpam-6015	266	29	i−n0	i−n0	X
ejpam-6015	266	30	+	+	SYM
ejpam-6015	266	31	1	1	NUM
ejpam-6015	266	32	→	→	SYM
ejpam-6015	266	33	0	0	NUM
ejpam-6015	266	34	as	as	ADP
ejpam-6015	266	35	n→	n→	ADV
ejpam-6015	266	36	+	+	PROPN
ejpam-6015	266	37	∞.	∞.	PROPN
ejpam-6015	266	38	thus	thus	ADV
ejpam-6015	266	39	,	,	PUNCT
ejpam-6015	266	40	lim	lim	PROPN
ejpam-6015	266	41	n	n	CCONJ
ejpam-6015	266	42	,	,	PUNCT
ejpam-6015	266	43	m→+∞	m→+∞	PROPN
ejpam-6015	266	44	d(ςn	d(ςn	NOUN
ejpam-6015	266	45	,	,	PUNCT
ejpam-6015	266	46	ςm	ςm	NOUN
ejpam-6015	266	47	)	)	PUNCT
ejpam-6015	266	48	=	=	SYM
ejpam-6015	266	49	0	0	X
ejpam-6015	266	50	.	.	PUNCT
ejpam-6015	267	1	so	so	ADV
ejpam-6015	267	2	{	{	PUNCT
ejpam-6015	267	3	ςn	ςn	NOUN
ejpam-6015	267	4	}	}	PUNCT
ejpam-6015	267	5	is	be	AUX
ejpam-6015	267	6	a	a	DET
ejpam-6015	267	7	cauchy	cauchy	ADJ
ejpam-6015	267	8	sequence	sequence	NOUN
ejpam-6015	267	9	in	in	ADP
ejpam-6015	267	10	the	the	DET
ejpam-6015	267	11	complete	complete	ADJ
ejpam-6015	267	12	ms	ms	NOUN
ejpam-6015	267	13	(	(	PUNCT
ejpam-6015	267	14	x	x	NOUN
ejpam-6015	267	15	,	,	PUNCT
ejpam-6015	267	16	d	d	NOUN
ejpam-6015	267	17	)	)	PUNCT
ejpam-6015	267	18	.	.	PUNCT
ejpam-6015	268	1	then	then	ADV
ejpam-6015	268	2	,	,	PUNCT
ejpam-6015	268	3	there	there	PRON
ejpam-6015	268	4	is	be	VERB
ejpam-6015	268	5	u	u	NOUN
ejpam-6015	268	6	∈	∈	PROPN
ejpam-6015	268	7	x	x	PUNCT
ejpam-6015	268	8	so	so	SCONJ
ejpam-6015	269	1	that	that	SCONJ
ejpam-6015	269	2	lim	lim	PROPN
ejpam-6015	269	3	n→+∞	n→+∞	PROPN
ejpam-6015	269	4	d(ςn	d(ςn	PROPN
ejpam-6015	269	5	,	,	PUNCT
ejpam-6015	269	6	u	u	NOUN
ejpam-6015	269	7	)	)	PUNCT
ejpam-6015	269	8	=	=	SYM
ejpam-6015	269	9	0	0	X
ejpam-6015	269	10	.	.	PUNCT
ejpam-6015	270	1	by	by	ADP
ejpam-6015	270	2	the	the	DET
ejpam-6015	270	3	lower	low	ADJ
ejpam-6015	270	4	semi	semi	NOUN
ejpam-6015	270	5	-	-	NOUN
ejpam-6015	270	6	continuity	continuity	NOUN
ejpam-6015	270	7	of	of	ADP
ejpam-6015	270	8	t	t	PROPN
ejpam-6015	270	9	,	,	PUNCT
ejpam-6015	270	10	we	we	PRON
ejpam-6015	270	11	obtain	obtain	VERB
ejpam-6015	270	12	that	that	SCONJ
ejpam-6015	270	13	u	u	PROPN
ejpam-6015	270	14	∈	∈	PROPN
ejpam-6015	270	15	tu	tu	PROPN
ejpam-6015	270	16	.	.	PROPN
ejpam-6015	270	17	remark	remark	PROPN
ejpam-6015	270	18	1	1	NUM
ejpam-6015	270	19	.	.	PUNCT
ejpam-6015	270	20	from	from	ADP
ejpam-6015	270	21	theorem	theorem	ADJ
ejpam-6015	270	22	4	4	NUM
ejpam-6015	270	23	,	,	PUNCT
ejpam-6015	270	24	several	several	ADJ
ejpam-6015	270	25	corollaries	corollary	NOUN
ejpam-6015	270	26	could	could	AUX
ejpam-6015	270	27	be	be	AUX
ejpam-6015	270	28	derived	derive	VERB
ejpam-6015	270	29	following	follow	VERB
ejpam-6015	270	30	particular	particular	ADJ
ejpam-6015	270	31	cases	case	NOUN
ejpam-6015	270	32	of	of	ADP
ejpam-6015	270	33	manageable	manageable	ADJ
ejpam-6015	270	34	functions	function	NOUN
ejpam-6015	270	35	.	.	PUNCT
ejpam-6015	271	1	the	the	DET
ejpam-6015	271	2	next	next	ADJ
ejpam-6015	271	3	example	example	NOUN
ejpam-6015	271	4	inspired	inspire	VERB
ejpam-6015	271	5	from	from	ADP
ejpam-6015	271	6	example	example	NOUN
ejpam-6015	271	7	3.3	3.3	NUM
ejpam-6015	271	8	in	in	ADP
ejpam-6015	271	9	[	[	X
ejpam-6015	271	10	1	1	NUM
ejpam-6015	271	11	]	]	PUNCT
ejpam-6015	271	12	makes	make	VERB
ejpam-6015	271	13	effective	effective	ADJ
ejpam-6015	271	14	theorem	theorem	NOUN
ejpam-6015	271	15	4	4	NUM
ejpam-6015	271	16	.	.	PUNCT
ejpam-6015	272	1	here	here	ADV
ejpam-6015	272	2	,	,	PUNCT
ejpam-6015	272	3	the	the	DET
ejpam-6015	272	4	theorem	theorem	NOUN
ejpam-6015	272	5	of	of	ADP
ejpam-6015	272	6	nadler	nadler	PROPN
ejpam-6015	273	1	[	[	X
ejpam-6015	273	2	10	10	NUM
ejpam-6015	273	3	]	]	PUNCT
ejpam-6015	273	4	is	be	AUX
ejpam-6015	273	5	not	not	PART
ejpam-6015	273	6	applicable	applicable	ADJ
ejpam-6015	273	7	..	..	PUNCT
ejpam-6015	273	8	example	example	NOUN
ejpam-6015	274	1	5	5	X
ejpam-6015	274	2	.	.	PUNCT
ejpam-6015	274	3	let	let	VERB
ejpam-6015	274	4	x	x	PUNCT
ejpam-6015	274	5	=	=	PRON
ejpam-6015	274	6	{	{	PUNCT
ejpam-6015	274	7	1	1	NUM
ejpam-6015	274	8	,	,	PUNCT
ejpam-6015	274	9	2	2	NUM
ejpam-6015	274	10	,	,	PUNCT
ejpam-6015	274	11	3	3	NUM
ejpam-6015	274	12	}	}	PUNCT
ejpam-6015	274	13	.	.	PUNCT
ejpam-6015	275	1	take	take	VERB
ejpam-6015	275	2	d	d	NOUN
ejpam-6015	275	3	the	the	DET
ejpam-6015	275	4	metric	metric	NOUN
ejpam-6015	275	5	on	on	ADP
ejpam-6015	275	6	x	x	PUNCT
ejpam-6015	275	7	given	give	VERB
ejpam-6015	275	8	as	as	ADP
ejpam-6015	275	9	d(ϖ	d(ϖ	PROPN
ejpam-6015	275	10	,	,	PUNCT
ejpam-6015	275	11	ς	ς	NOUN
ejpam-6015	275	12	)	)	PUNCT
ejpam-6015	275	13	=	=	NOUN
ejpam-6015	275	14	d(ς,ϖ	d(ς,ϖ	NOUN
ejpam-6015	275	15	)	)	PUNCT
ejpam-6015	275	16	,	,	PUNCT
ejpam-6015	275	17	d(ς	d(ς	PROPN
ejpam-6015	275	18	,	,	PUNCT
ejpam-6015	275	19	ς	ς	NOUN
ejpam-6015	275	20	)	)	PUNCT
ejpam-6015	275	21	=	=	SYM
ejpam-6015	275	22	0	0	NUM
ejpam-6015	276	1	∀ϖ	∀ϖ	PROPN
ejpam-6015	276	2	,	,	PUNCT
ejpam-6015	276	3	ς	ς	PROPN
ejpam-6015	276	4	∈	∈	PROPN
ejpam-6015	276	5	x	x	X
ejpam-6015	276	6	and	and	CCONJ
ejpam-6015	276	7	d(1	d(1	PROPN
ejpam-6015	276	8	,	,	PUNCT
ejpam-6015	276	9	2	2	NUM
ejpam-6015	276	10	)	)	PUNCT
ejpam-6015	276	11	=	=	SYM
ejpam-6015	276	12	1	1	X
ejpam-6015	276	13	,	,	PUNCT
ejpam-6015	276	14	d(1	d(1	NOUN
ejpam-6015	276	15	,	,	PUNCT
ejpam-6015	276	16	3	3	X
ejpam-6015	276	17	)	)	PUNCT
ejpam-6015	276	18	=	=	SYM
ejpam-6015	276	19	4	4	X
ejpam-6015	276	20	,	,	PUNCT
ejpam-6015	276	21	d(2	d(2	PROPN
ejpam-6015	276	22	,	,	PUNCT
ejpam-6015	276	23	3	3	NUM
ejpam-6015	276	24	)	)	PUNCT
ejpam-6015	276	25	=	=	SYM
ejpam-6015	276	26	5	5	X
ejpam-6015	276	27	.	.	X
ejpam-6015	276	28	notice	notice	VERB
ejpam-6015	276	29	that	that	SCONJ
ejpam-6015	276	30	(	(	PUNCT
ejpam-6015	276	31	x	x	X
ejpam-6015	276	32	,	,	PUNCT
ejpam-6015	276	33	d	d	NOUN
ejpam-6015	276	34	)	)	PUNCT
ejpam-6015	276	35	is	be	AUX
ejpam-6015	276	36	a	a	DET
ejpam-6015	276	37	complete	complete	ADJ
ejpam-6015	276	38	ms	ms	PROPN
ejpam-6015	276	39	.	.	PROPN
ejpam-6015	276	40	choose	choose	PROPN
ejpam-6015	276	41	t	t	PROPN
ejpam-6015	276	42	:	:	PUNCT
ejpam-6015	276	43	x	x	X
ejpam-6015	276	44	→	→	X
ejpam-6015	276	45	cb(x	cb(x	NUM
ejpam-6015	276	46	)	)	PUNCT
ejpam-6015	276	47	as	as	ADP
ejpam-6015	276	48	t1	t1	NOUN
ejpam-6015	276	49	=	=	SYM
ejpam-6015	276	50	t3	t3	PROPN
ejpam-6015	276	51	=	=	PUNCT
ejpam-6015	276	52	{	{	PUNCT
ejpam-6015	276	53	1	1	NUM
ejpam-6015	276	54	}	}	PUNCT
ejpam-6015	276	55	and	and	CCONJ
ejpam-6015	276	56	t2	t2	PROPN
ejpam-6015	276	57	=	=	SYM
ejpam-6015	276	58	{	{	PUNCT
ejpam-6015	276	59	1	1	NUM
ejpam-6015	276	60	,	,	PUNCT
ejpam-6015	276	61	3	3	NUM
ejpam-6015	276	62	}	}	PUNCT
ejpam-6015	276	63	.	.	PUNCT
ejpam-6015	277	1	h.	h.	PROPN
ejpam-6015	277	2	aydi	aydi	VERB
ejpam-6015	277	3	et	et	PROPN
ejpam-6015	277	4	al	al	PROPN
ejpam-6015	277	5	.	.	PUNCT
ejpam-6015	277	6	/	/	SYM
ejpam-6015	277	7	eur	eur	PROPN
ejpam-6015	277	8	.	.	PUNCT
ejpam-6015	278	1	j.	j.	PROPN
ejpam-6015	278	2	pure	pure	PROPN
ejpam-6015	278	3	appl	appl	PROPN
ejpam-6015	278	4	.	.	PROPN
ejpam-6015	278	5	math	math	PROPN
ejpam-6015	278	6	,	,	PUNCT
ejpam-6015	278	7	18	18	NUM
ejpam-6015	278	8	(	(	PUNCT
ejpam-6015	278	9	2	2	NUM
ejpam-6015	278	10	)	)	PUNCT
ejpam-6015	278	11	(	(	PUNCT
ejpam-6015	278	12	2025	2025	NUM
ejpam-6015	278	13	)	)	PUNCT
ejpam-6015	278	14	,	,	PUNCT
ejpam-6015	278	15	6015	6015	NUM
ejpam-6015	278	16	12	12	NUM
ejpam-6015	278	17	of	of	ADP
ejpam-6015	278	18	14	14	NUM
ejpam-6015	278	19	we	we	PRON
ejpam-6015	278	20	point	point	VERB
ejpam-6015	278	21	out	out	ADP
ejpam-6015	278	22	that	that	SCONJ
ejpam-6015	278	23	t	t	PROPN
ejpam-6015	278	24	is	be	AUX
ejpam-6015	278	25	not	not	PART
ejpam-6015	278	26	a	a	DET
ejpam-6015	278	27	contraction	contraction	NOUN
ejpam-6015	278	28	in	in	ADP
ejpam-6015	278	29	the	the	DET
ejpam-6015	278	30	sense	sense	NOUN
ejpam-6015	278	31	of	of	ADP
ejpam-6015	278	32	nadler	nadler	NOUN
ejpam-6015	279	1	[	[	X
ejpam-6015	279	2	10	10	NUM
ejpam-6015	279	3	]	]	PUNCT
ejpam-6015	279	4	.	.	PUNCT
ejpam-6015	280	1	indeed	indeed	ADV
ejpam-6015	280	2	,	,	PUNCT
ejpam-6015	280	3	h(t1	h(t1	PROPN
ejpam-6015	280	4	,	,	PUNCT
ejpam-6015	280	5	t2	t2	NOUN
ejpam-6015	280	6	)	)	PUNCT
ejpam-6015	280	7	=	=	SYM
ejpam-6015	280	8	max{d(1	max{d(1	NOUN
ejpam-6015	280	9	,	,	PUNCT
ejpam-6015	280	10	1	1	NUM
ejpam-6015	280	11	)	)	PUNCT
ejpam-6015	280	12	,	,	PUNCT
ejpam-6015	280	13	d(1	d(1	PROPN
ejpam-6015	280	14	,	,	PUNCT
ejpam-6015	280	15	3	3	NUM
ejpam-6015	280	16	)	)	PUNCT
ejpam-6015	280	17	}	}	PUNCT
ejpam-6015	280	18	=	=	SYM
ejpam-6015	280	19	4	4	NUM
ejpam-6015	280	20	>	>	SYM
ejpam-6015	280	21	1	1	NUM
ejpam-6015	280	22	=	=	SYM
ejpam-6015	280	23	d(1	d(1	NOUN
ejpam-6015	280	24	,	,	PUNCT
ejpam-6015	280	25	2	2	NUM
ejpam-6015	280	26	)	)	PUNCT
ejpam-6015	280	27	.	.	PUNCT
ejpam-6015	281	1	we	we	PRON
ejpam-6015	281	2	now	now	ADV
ejpam-6015	281	3	introduce	introduce	VERB
ejpam-6015	281	4	the	the	DET
ejpam-6015	281	5	mapping	mapping	NOUN
ejpam-6015	281	6	θ	θ	NOUN
ejpam-6015	281	7	:	:	PUNCT
ejpam-6015	282	1	[	[	X
ejpam-6015	282	2	0,+∞	0,+∞	NUM
ejpam-6015	282	3	)	)	PUNCT
ejpam-6015	282	4	→	→	PUNCT
ejpam-6015	283	1	[	[	X
ejpam-6015	283	2	0,+∞	0,+∞	NUM
ejpam-6015	283	3	)	)	PUNCT
ejpam-6015	283	4	defined	define	VERB
ejpam-6015	283	5	by	by	ADP
ejpam-6015	283	6	θ(t	θ(t	NOUN
ejpam-6015	283	7	)	)	PUNCT
ejpam-6015	283	8	=	=	PUNCT
ejpam-6015	283	9			NUM
ejpam-6015	283	10	7	7	NUM
ejpam-6015	283	11	t	t	NOUN
ejpam-6015	283	12	sinh	sinh	NOUN
ejpam-6015	283	13	1	1	NUM
ejpam-6015	283	14	if	if	SCONJ
ejpam-6015	283	15	0	0	NUM
ejpam-6015	283	16	≤	≤	NUM
ejpam-6015	283	17	t	t	PROPN
ejpam-6015	283	18	≤	≤	PROPN
ejpam-6015	283	19	sinh	sinh	NOUN
ejpam-6015	283	20	1	1	NUM
ejpam-6015	283	21	,	,	PUNCT
ejpam-6015	283	22	2	2	NUM
ejpam-6015	283	23	t	t	NOUN
ejpam-6015	283	24	sinh	sinh	NOUN
ejpam-6015	283	25	4	4	NUM
ejpam-6015	283	26	if	if	SCONJ
ejpam-6015	283	27	sinh	sinh	VERB
ejpam-6015	283	28	1	1	NUM
ejpam-6015	283	29	<	<	X
ejpam-6015	283	30	t	t	X
ejpam-6015	283	31	<	<	X
ejpam-6015	283	32	sinh	sinh	PROPN
ejpam-6015	283	33	4	4	NUM
ejpam-6015	283	34	,	,	PUNCT
ejpam-6015	283	35	5	5	NUM
ejpam-6015	283	36	t	t	NOUN
ejpam-6015	283	37	4	4	NUM
ejpam-6015	283	38	sinh	sinh	NOUN
ejpam-6015	283	39	5	5	NUM
ejpam-6015	283	40	if	if	SCONJ
ejpam-6015	283	41	t	t	PROPN
ejpam-6015	283	42	≥	≥	NOUN
ejpam-6015	283	43	sinh	sinh	VERB
ejpam-6015	283	44	4	4	NUM
ejpam-6015	283	45	.	.	PUNCT
ejpam-6015	283	46	clearly	clearly	ADV
ejpam-6015	283	47	,	,	PUNCT
ejpam-6015	283	48	θ(t	θ(t	PROPN
ejpam-6015	283	49	)	)	PUNCT
ejpam-6015	283	50	≥	≥	NOUN
ejpam-6015	283	51	2	2	NUM
ejpam-6015	283	52	sinh	sinh	NOUN
ejpam-6015	283	53	5	5	NUM
ejpam-6015	283	54	t	t	PROPN
ejpam-6015	283	55	,	,	PUNCT
ejpam-6015	283	56	∀t	∀t	PROPN
ejpam-6015	283	57	≥	≥	NOUN
ejpam-6015	283	58	0	0	NUM
ejpam-6015	283	59	,	,	PUNCT
ejpam-6015	283	60	which	which	PRON
ejpam-6015	283	61	shows	show	VERB
ejpam-6015	283	62	that	that	SCONJ
ejpam-6015	283	63	θ	θ	PROPN
ejpam-6015	283	64	∈	∈	PROPN
ejpam-6015	283	65	θ1	θ1	NOUN
ejpam-6015	283	66	and	and	CCONJ
ejpam-6015	283	67	θ(0	θ(0	PROPN
ejpam-6015	283	68	)	)	PUNCT
ejpam-6015	283	69	=	=	SYM
ejpam-6015	284	1	0	0	X
ejpam-6015	284	2	.	.	PUNCT
ejpam-6015	285	1	furthermore	furthermore	ADV
ejpam-6015	285	2	,	,	PUNCT
ejpam-6015	285	3	take	take	VERB
ejpam-6015	285	4	η(t	η(t	NOUN
ejpam-6015	285	5	,	,	PUNCT
ejpam-6015	285	6	s	s	PART
ejpam-6015	285	7	)	)	PUNCT
ejpam-6015	285	8	=	=	SYM
ejpam-6015	285	9	ks−	ks−	PROPN
ejpam-6015	285	10	t	t	NOUN
ejpam-6015	285	11	for	for	ADP
ejpam-6015	285	12	all	all	DET
ejpam-6015	285	13	s	s	PROPN
ejpam-6015	285	14	,	,	PUNCT
ejpam-6015	285	15	t	t	PROPN
ejpam-6015	285	16	∈	∈	PROPN
ejpam-6015	285	17	r	r	NOUN
ejpam-6015	285	18	with	with	ADP
ejpam-6015	285	19	k	k	PROPN
ejpam-6015	285	20	∈	∈	PROPN
ejpam-6015	285	21	[	[	X
ejpam-6015	285	22	25	25	NUM
ejpam-6015	285	23	,	,	PUNCT
ejpam-6015	285	24	1	1	NUM
ejpam-6015	285	25	)	)	PUNCT
ejpam-6015	285	26	.	.	PUNCT
ejpam-6015	286	1	we	we	PRON
ejpam-6015	286	2	have	have	VERB
ejpam-6015	286	3	hθ(t1	hθ(t1	NUM
ejpam-6015	286	4	,	,	PUNCT
ejpam-6015	286	5	t2	t2	NOUN
ejpam-6015	286	6	)	)	PUNCT
ejpam-6015	286	7	=	=	SYM
ejpam-6015	286	8	max{dθ(1	max{dθ(1	PROPN
ejpam-6015	286	9	,	,	PUNCT
ejpam-6015	286	10	3	3	NUM
ejpam-6015	286	11	)	)	PUNCT
ejpam-6015	286	12	,	,	PUNCT
ejpam-6015	286	13	dθ(1	dθ(1	PROPN
ejpam-6015	286	14	,	,	PUNCT
ejpam-6015	286	15	1	1	NUM
ejpam-6015	286	16	)	)	PUNCT
ejpam-6015	286	17	}	}	PUNCT
ejpam-6015	287	1	=	=	SYM
ejpam-6015	287	2	dθ(1	dθ(1	NOUN
ejpam-6015	287	3	,	,	PUNCT
ejpam-6015	287	4	3	3	NUM
ejpam-6015	287	5	)	)	PUNCT
ejpam-6015	287	6	=	=	NOUN
ejpam-6015	287	7	θ(sinh	θ(sinh	NOUN
ejpam-6015	287	8	4	4	NUM
ejpam-6015	287	9	)	)	PUNCT
ejpam-6015	287	10	=	=	SYM
ejpam-6015	287	11	2	2	NUM
ejpam-6015	287	12	=	=	SYM
ejpam-6015	287	13	2	2	NUM
ejpam-6015	287	14	7	7	NUM
ejpam-6015	287	15	×	×	NOUN
ejpam-6015	287	16	7	7	NUM
ejpam-6015	287	17	=	=	SYM
ejpam-6015	287	18	2	2	NUM
ejpam-6015	287	19	7	7	NUM
ejpam-6015	287	20	×	×	NOUN
ejpam-6015	287	21	θ(sinh	θ(sinh	NOUN
ejpam-6015	287	22	1	1	NUM
ejpam-6015	287	23	)	)	PUNCT
ejpam-6015	287	24	=	=	SYM
ejpam-6015	287	25	2	2	NUM
ejpam-6015	287	26	7	7	NUM
ejpam-6015	287	27	dθ(1	dθ(1	NOUN
ejpam-6015	287	28	,	,	PUNCT
ejpam-6015	287	29	2	2	NUM
ejpam-6015	287	30	)	)	PUNCT
ejpam-6015	287	31	≤	≤	NUM
ejpam-6015	287	32	2	2	NUM
ejpam-6015	287	33	5	5	NUM
ejpam-6015	287	34	dθ(1	dθ(1	NOUN
ejpam-6015	287	35	,	,	PUNCT
ejpam-6015	287	36	2	2	NUM
ejpam-6015	287	37	)	)	PUNCT
ejpam-6015	287	38	.	.	PUNCT
ejpam-6015	288	1	also	also	ADV
ejpam-6015	288	2	,	,	PUNCT
ejpam-6015	288	3	hθ(t2	hθ(t2	NOUN
ejpam-6015	288	4	,	,	PUNCT
ejpam-6015	288	5	t3	t3	NOUN
ejpam-6015	288	6	)	)	PUNCT
ejpam-6015	288	7	=	=	SYM
ejpam-6015	288	8	2	2	NUM
ejpam-6015	288	9	=	=	SYM
ejpam-6015	288	10	2	2	NUM
ejpam-6015	288	11	5	5	NUM
ejpam-6015	288	12	×	×	NOUN
ejpam-6015	288	13	5	5	NUM
ejpam-6015	288	14	=	=	SYM
ejpam-6015	288	15	2	2	NUM
ejpam-6015	288	16	5	5	NUM
ejpam-6015	288	17	θ(sinh	θ(sinh	NOUN
ejpam-6015	288	18	5	5	NUM
ejpam-6015	288	19	)	)	PUNCT
ejpam-6015	288	20	=	=	SYM
ejpam-6015	288	21	2	2	NUM
ejpam-6015	288	22	5	5	NUM
ejpam-6015	288	23	dθ(2	dθ(2	PROPN
ejpam-6015	288	24	,	,	PUNCT
ejpam-6015	288	25	3	3	NUM
ejpam-6015	288	26	)	)	PUNCT
ejpam-6015	288	27	.	.	PUNCT
ejpam-6015	289	1	furthermore	furthermore	ADV
ejpam-6015	289	2	,	,	PUNCT
ejpam-6015	289	3	hθ(t1	hθ(t1	INTJ
ejpam-6015	289	4	,	,	PUNCT
ejpam-6015	289	5	t3	t3	PROPN
ejpam-6015	289	6	)	)	PUNCT
ejpam-6015	289	7	=	=	SYM
ejpam-6015	289	8	0	0	NUM
ejpam-6015	289	9	≤	≤	NUM
ejpam-6015	289	10	2	2	NUM
ejpam-6015	289	11	5	5	NUM
ejpam-6015	289	12	dθ(1	dθ(1	NOUN
ejpam-6015	289	13	,	,	PUNCT
ejpam-6015	289	14	3	3	NUM
ejpam-6015	289	15	)	)	PUNCT
ejpam-6015	289	16	,	,	PUNCT
ejpam-6015	289	17	which	which	PRON
ejpam-6015	289	18	implies	imply	VERB
ejpam-6015	289	19	hθ(tς	hθ(tς	PROPN
ejpam-6015	289	20	,	,	PUNCT
ejpam-6015	289	21	tϖ	tϖ	NOUN
ejpam-6015	289	22	)	)	PUNCT
ejpam-6015	289	23	≤	≤	NOUN
ejpam-6015	289	24	2	2	NUM
ejpam-6015	289	25	5	5	NUM
ejpam-6015	289	26	dθ(ϖ	dθ(ϖ	NOUN
ejpam-6015	289	27	,	,	PUNCT
ejpam-6015	289	28	ς)∀ϖ	ς)∀ϖ	ADJ
ejpam-6015	289	29	,	,	PUNCT
ejpam-6015	289	30	ς	ς	PROPN
ejpam-6015	289	31	∈	∈	PROPN
ejpam-6015	289	32	x.	x.	NOUN
ejpam-6015	290	1	we	we	PRON
ejpam-6015	290	2	also	also	ADV
ejpam-6015	290	3	have	have	VERB
ejpam-6015	290	4	η(hθ(tς	η(hθ(tς	NOUN
ejpam-6015	290	5	,	,	PUNCT
ejpam-6015	290	6	tϖ	tϖ	NOUN
ejpam-6015	290	7	)	)	PUNCT
ejpam-6015	290	8	,	,	PUNCT
ejpam-6015	290	9	dθ(ϖ	dθ(ϖ	PROPN
ejpam-6015	290	10	,	,	PUNCT
ejpam-6015	290	11	ς	ς	NOUN
ejpam-6015	290	12	)	)	PUNCT
ejpam-6015	290	13	)	)	PUNCT
ejpam-6015	291	1	=	=	SYM
ejpam-6015	291	2	khθ(tς	khθ(tς	PROPN
ejpam-6015	291	3	,	,	PUNCT
ejpam-6015	291	4	tϖ)−	tϖ)−	NUM
ejpam-6015	291	5	dθ(ϖ	dθ(ϖ	PUNCT
ejpam-6015	291	6	,	,	PUNCT
ejpam-6015	291	7	ς	ς	PROPN
ejpam-6015	291	8	)	)	PUNCT
ejpam-6015	291	9	≥	≥	NOUN
ejpam-6015	291	10	(	(	PUNCT
ejpam-6015	291	11	k	k	NOUN
ejpam-6015	291	12	−	−	PROPN
ejpam-6015	291	13	2	2	NUM
ejpam-6015	291	14	5	5	NUM
ejpam-6015	291	15	)	)	PUNCT
ejpam-6015	291	16	dθ(ϖ	dθ(ϖ	PROPN
ejpam-6015	291	17	,	,	PUNCT
ejpam-6015	291	18	ς	ς	PROPN
ejpam-6015	291	19	)	)	PUNCT
ejpam-6015	291	20	≥	≥	NOUN
ejpam-6015	291	21	0∀ϖ	0∀ϖ	NOUN
ejpam-6015	291	22	,	,	PUNCT
ejpam-6015	291	23	ς	ς	PROPN
ejpam-6015	291	24	∈	∈	PROPN
ejpam-6015	291	25	x.	x.	NOUN
ejpam-6015	291	26	let	let	VERB
ejpam-6015	291	27	ς	ς	PROPN
ejpam-6015	291	28	∈	∈	PROPN
ejpam-6015	291	29	x	x	X
ejpam-6015	291	30	and	and	CCONJ
ejpam-6015	291	31	{	{	PUNCT
ejpam-6015	291	32	ςn	ςn	NOUN
ejpam-6015	291	33	}	}	PUNCT
ejpam-6015	291	34	⊂	⊂	X
ejpam-6015	291	35	x	x	PUNCT
ejpam-6015	292	1	so	so	ADV
ejpam-6015	292	2	that	that	PRON
ejpam-6015	292	3	limn→+∞	limn→+∞	VERB
ejpam-6015	292	4	d(ςn	d(ςn	NOUN
ejpam-6015	292	5	,	,	PUNCT
ejpam-6015	292	6	ς	ς	NOUN
ejpam-6015	292	7	)	)	PUNCT
ejpam-6015	292	8	=	=	SYM
ejpam-6015	293	1	0	0	X
ejpam-6015	293	2	.	.	PUNCT
ejpam-6015	294	1	then	then	ADV
ejpam-6015	294	2	,	,	PUNCT
ejpam-6015	294	3	there	there	PRON
ejpam-6015	294	4	is	be	VERB
ejpam-6015	294	5	n0	n0	NUM
ejpam-6015	294	6	≥	≥	NOUN
ejpam-6015	294	7	0	0	PUNCT
ejpam-6015	295	1	so	so	SCONJ
ejpam-6015	295	2	that	that	SCONJ
ejpam-6015	295	3	ςn	ςn	PRON
ejpam-6015	295	4	=	=	PUNCT
ejpam-6015	295	5	ς	ς	PROPN
ejpam-6015	295	6	for	for	ADP
ejpam-6015	295	7	all	all	DET
ejpam-6015	295	8	n0	n0	NUM
ejpam-6015	295	9	≥	≥	NOUN
ejpam-6015	295	10	0	0	NUM
ejpam-6015	295	11	.	.	PUNCT
ejpam-6015	296	1	then	then	ADV
ejpam-6015	296	2	tςn	tςn	ADV
ejpam-6015	296	3	=	=	SYM
ejpam-6015	296	4	tς	tς	PROPN
ejpam-6015	296	5	for	for	ADP
ejpam-6015	296	6	all	all	DET
ejpam-6015	296	7	n0	n0	PROPN
ejpam-6015	296	8	≥	≥	NOUN
ejpam-6015	296	9	0	0	NUM
ejpam-6015	296	10	.	.	PUNCT
ejpam-6015	297	1	henc	henc	PROPN
ejpam-6015	297	2	,	,	PUNCT
ejpam-6015	297	3	e	e	X
ejpam-6015	297	4	d(ς	d(ς	PROPN
ejpam-6015	297	5	,	,	PUNCT
ejpam-6015	297	6	t	t	PROPN
ejpam-6015	297	7	ς	ς	PROPN
ejpam-6015	297	8	)	)	PUNCT
ejpam-6015	297	9	=	=	SYM
ejpam-6015	298	1	d(ςn	d(ςn	NOUN
ejpam-6015	298	2	,	,	PUNCT
ejpam-6015	298	3	t	t	NOUN
ejpam-6015	298	4	ςn	ςn	NOUN
ejpam-6015	298	5	)	)	PUNCT
ejpam-6015	298	6	for	for	ADP
ejpam-6015	298	7	all	all	DET
ejpam-6015	298	8	n0	n0	PROPN
ejpam-6015	298	9	≥	≥	NOUN
ejpam-6015	298	10	0	0	NUM
ejpam-6015	298	11	.	.	PUNCT
ejpam-6015	299	1	finally	finally	ADV
ejpam-6015	299	2	,	,	PUNCT
ejpam-6015	299	3	we	we	PRON
ejpam-6015	299	4	get	get	VERB
ejpam-6015	299	5	d(ς	d(ς	PROPN
ejpam-6015	299	6	,	,	PUNCT
ejpam-6015	299	7	t	t	PROPN
ejpam-6015	299	8	ς	ς	PROPN
ejpam-6015	299	9	)	)	PUNCT
ejpam-6015	299	10	=	=	PROPN
ejpam-6015	299	11	lim	lim	PROPN
ejpam-6015	299	12	inf	inf	PROPN
ejpam-6015	299	13	n→+∞	n→+∞	PROPN
ejpam-6015	299	14	d(ςn	d(ςn	PROPN
ejpam-6015	299	15	,	,	PUNCT
ejpam-6015	299	16	t	t	PROPN
ejpam-6015	299	17	ςn	ςn	PROPN
ejpam-6015	299	18	)	)	PUNCT
ejpam-6015	299	19	.	.	PUNCT
ejpam-6015	300	1	consequently	consequently	ADV
ejpam-6015	300	2	,	,	PUNCT
ejpam-6015	300	3	all	all	PRON
ejpam-6015	300	4	required	require	VERB
ejpam-6015	300	5	hypothesises	hypothesis	NOUN
ejpam-6015	300	6	of	of	ADP
ejpam-6015	300	7	theorem	theorem	ADJ
ejpam-6015	300	8	4	4	NUM
ejpam-6015	300	9	hold	hold	NOUN
ejpam-6015	300	10	.	.	PUNCT
ejpam-6015	301	1	here	here	ADV
ejpam-6015	301	2	,	,	PUNCT
ejpam-6015	301	3	1	1	NUM
ejpam-6015	301	4	is	be	AUX
ejpam-6015	301	5	a	a	DET
ejpam-6015	301	6	fp	fp	NOUN
ejpam-6015	301	7	of	of	ADP
ejpam-6015	301	8	t	t	PROPN
ejpam-6015	301	9	.	.	PUNCT
ejpam-6015	302	1	h.	h.	PROPN
ejpam-6015	302	2	aydi	aydi	VERB
ejpam-6015	302	3	et	et	PROPN
ejpam-6015	302	4	al	al	PROPN
ejpam-6015	302	5	.	.	PUNCT
ejpam-6015	302	6	/	/	SYM
ejpam-6015	302	7	eur	eur	PROPN
ejpam-6015	302	8	.	.	PUNCT
ejpam-6015	303	1	j.	j.	PROPN
ejpam-6015	303	2	pure	pure	PROPN
ejpam-6015	303	3	appl	appl	PROPN
ejpam-6015	303	4	.	.	PROPN
ejpam-6015	303	5	math	math	PROPN
ejpam-6015	303	6	,	,	PUNCT
ejpam-6015	303	7	18	18	NUM
ejpam-6015	303	8	(	(	PUNCT
ejpam-6015	303	9	2	2	NUM
ejpam-6015	303	10	)	)	PUNCT
ejpam-6015	303	11	(	(	PUNCT
ejpam-6015	303	12	2025	2025	NUM
ejpam-6015	303	13	)	)	PUNCT
ejpam-6015	303	14	,	,	PUNCT
ejpam-6015	303	15	6015	6015	NUM
ejpam-6015	303	16	13	13	NUM
ejpam-6015	303	17	of	of	ADP
ejpam-6015	303	18	14	14	NUM
ejpam-6015	303	19	4	4	NUM
ejpam-6015	303	20	.	.	PUNCT
ejpam-6015	303	21	conclusion	conclusion	NOUN
ejpam-6015	303	22	in	in	ADP
ejpam-6015	303	23	this	this	DET
ejpam-6015	303	24	work	work	NOUN
ejpam-6015	303	25	,	,	PUNCT
ejpam-6015	303	26	we	we	PRON
ejpam-6015	303	27	initiated	initiate	VERB
ejpam-6015	303	28	the	the	DET
ejpam-6015	303	29	concept	concept	NOUN
ejpam-6015	303	30	of	of	ADP
ejpam-6015	303	31	a	a	DET
ejpam-6015	303	32	hausdorff	hausdorff	NOUN
ejpam-6015	303	33	θ	θ	ADJ
ejpam-6015	303	34	-	-	ADJ
ejpam-6015	303	35	hyperbolic	hyperbolic	ADJ
ejpam-6015	303	36	sine	sine	NOUN
ejpam-6015	303	37	distance	distance	NOUN
ejpam-6015	303	38	function	function	NOUN
ejpam-6015	303	39	.	.	PUNCT
ejpam-6015	304	1	we	we	PRON
ejpam-6015	304	2	proved	prove	VERB
ejpam-6015	304	3	two	two	NUM
ejpam-6015	304	4	fp	fp	NOUN
ejpam-6015	304	5	results	result	NOUN
ejpam-6015	304	6	for	for	ADP
ejpam-6015	304	7	multivalued	multivalued	ADJ
ejpam-6015	304	8	contraction	contraction	NOUN
ejpam-6015	304	9	mappings	mapping	NOUN
ejpam-6015	304	10	,	,	PUNCT
ejpam-6015	304	11	one	one	NUM
ejpam-6015	304	12	of	of	ADP
ejpam-6015	304	13	nadler	nadler	NOUN
ejpam-6015	304	14	type	type	NOUN
ejpam-6015	304	15	,	,	PUNCT
ejpam-6015	304	16	and	and	CCONJ
ejpam-6015	304	17	the	the	DET
ejpam-6015	304	18	second	second	ADJ
ejpam-6015	304	19	using	use	VERB
ejpam-6015	304	20	manageable	manageable	ADJ
ejpam-6015	304	21	functions	function	NOUN
ejpam-6015	304	22	.	.	PUNCT
ejpam-6015	305	1	as	as	ADP
ejpam-6015	305	2	open	open	ADJ
ejpam-6015	305	3	problems	problem	NOUN
ejpam-6015	305	4	,	,	PUNCT
ejpam-6015	305	5	we	we	PRON
ejpam-6015	305	6	suggest	suggest	VERB
ejpam-6015	305	7	to	to	PART
ejpam-6015	305	8	prove	prove	VERB
ejpam-6015	305	9	further	further	ADJ
ejpam-6015	305	10	fp	fp	X
ejpam-6015	305	11	results	result	NOUN
ejpam-6015	305	12	for	for	ADP
ejpam-6015	305	13	multivalued	multivalued	ADJ
ejpam-6015	305	14	mappings	mapping	NOUN
ejpam-6015	305	15	,	,	PUNCT
ejpam-6015	305	16	using	use	VERB
ejpam-6015	305	17	either	either	CCONJ
ejpam-6015	305	18	different	different	ADJ
ejpam-6015	305	19	types	type	NOUN
ejpam-6015	305	20	of	of	ADP
ejpam-6015	305	21	control	control	NOUN
ejpam-6015	305	22	function	function	NOUN
ejpam-6015	305	23	,	,	PUNCT
ejpam-6015	305	24	like	like	ADP
ejpam-6015	305	25	:	:	PUNCT
ejpam-6015	305	26	(	(	PUNCT
ejpam-6015	305	27	i	i	NOUN
ejpam-6015	305	28	)	)	PUNCT
ejpam-6015	305	29	implict	implict	ADJ
ejpam-6015	305	30	functions	function	NOUN
ejpam-6015	305	31	;	;	PUNCT
ejpam-6015	305	32	(	(	PUNCT
ejpam-6015	305	33	ii	ii	NOUN
ejpam-6015	305	34	)	)	PUNCT
ejpam-6015	305	35	α	α	NOUN
ejpam-6015	305	36	-	-	NOUN
ejpam-6015	305	37	admissibility	admissibility	NOUN
ejpam-6015	305	38	,	,	PUNCT
ejpam-6015	305	39	or	or	CCONJ
ejpam-6015	305	40	,	,	PUNCT
ejpam-6015	305	41	via	via	ADP
ejpam-6015	305	42	generalized	generalized	ADJ
ejpam-6015	305	43	metrics	metric	NOUN
ejpam-6015	305	44	.	.	PUNCT
ejpam-6015	306	1	acknowledgements	acknowledgement	VERB
ejpam-6015	306	2	the	the	DET
ejpam-6015	306	3	authors	author	NOUN
ejpam-6015	306	4	i.	i.	PROPN
ejpam-6015	306	5	ayoob	ayoob	PROPN
ejpam-6015	306	6	and	and	CCONJ
ejpam-6015	306	7	n.	n.	PROPN
ejpam-6015	306	8	mlaiki	mlaiki	PROPN
ejpam-6015	306	9	would	would	AUX
ejpam-6015	306	10	like	like	VERB
ejpam-6015	306	11	to	to	PART
ejpam-6015	306	12	thank	thank	VERB
ejpam-6015	306	13	prince	prince	PROPN
ejpam-6015	306	14	sultan	sultan	PROPN
ejpam-6015	306	15	university	university	PROPN
ejpam-6015	306	16	for	for	ADP
ejpam-6015	306	17	paying	pay	VERB
ejpam-6015	306	18	the	the	DET
ejpam-6015	306	19	apc	apc	PROPN
ejpam-6015	306	20	through	through	ADP
ejpam-6015	306	21	tas	ta	NOUN
ejpam-6015	306	22	lab	lab	PROPN
ejpam-6015	306	23	.	.	PUNCT
ejpam-6015	307	1	the	the	DET
ejpam-6015	307	2	authors	author	NOUN
ejpam-6015	307	3	declare	declare	VERB
ejpam-6015	307	4	no	no	DET
ejpam-6015	307	5	conflict	conflict	NOUN
ejpam-6015	307	6	of	of	ADP
ejpam-6015	307	7	interest	interest	NOUN
ejpam-6015	307	8	references	reference	NOUN
ejpam-6015	307	9	[	[	X
ejpam-6015	307	10	1	1	NUM
ejpam-6015	307	11	]	]	PUNCT
ejpam-6015	307	12	b.	b.	PROPN
ejpam-6015	307	13	samet	samet	PROPN
ejpam-6015	307	14	m.	m.	PROPN
ejpam-6015	307	15	jleli	jleli	PROPN
ejpam-6015	307	16	.	.	PUNCT
ejpam-6015	308	1	on	on	ADP
ejpam-6015	308	2	θ	θ	ADJ
ejpam-6015	308	3	-	-	ADJ
ejpam-6015	308	4	hyperbolic	hyperbolic	ADJ
ejpam-6015	308	5	sine	sine	NOUN
ejpam-6015	308	6	distance	distance	NOUN
ejpam-6015	308	7	functions	function	NOUN
ejpam-6015	308	8	and	and	CCONJ
ejpam-6015	308	9	existence	existence	NOUN
ejpam-6015	308	10	results	result	VERB
ejpam-6015	308	11	in	in	ADP
ejpam-6015	308	12	complete	complete	ADJ
ejpam-6015	308	13	metric	metric	ADJ
ejpam-6015	308	14	spaces	space	NOUN
ejpam-6015	308	15	.	.	PUNCT
ejpam-6015	309	1	aims	aim	VERB
ejpam-6015	309	2	mathematics	mathematic	NOUN
ejpam-6015	309	3	,	,	PUNCT
ejpam-6015	309	4	9:29001–29017	9:29001–29017	NUM
ejpam-6015	309	5	,	,	PUNCT
ejpam-6015	309	6	2024	2024	NUM
ejpam-6015	309	7	.	.	PUNCT
ejpam-6015	310	1	[	[	X
ejpam-6015	310	2	2	2	NUM
ejpam-6015	310	3	]	]	PUNCT
ejpam-6015	310	4	m.	m.	NOUN
ejpam-6015	310	5	fréchet	fréchet	PROPN
ejpam-6015	310	6	.	.	PUNCT
ejpam-6015	311	1	sur	sur	PROPN
ejpam-6015	311	2	quelques	quelques	PROPN
ejpam-6015	311	3	points	point	NOUN
ejpam-6015	311	4	du	du	PROPN
ejpam-6015	311	5	calcul	calcul	PROPN
ejpam-6015	311	6	fonctionnel	fonctionnel	PROPN
ejpam-6015	311	7	.	.	PUNCT
ejpam-6015	312	1	rend	rend	VERB
ejpam-6015	312	2	.	.	PUNCT
ejpam-6015	313	1	circ	circ	PROPN
ejpam-6015	313	2	.	.	PUNCT
ejpam-6015	314	1	mat	mat	NOUN
ejpam-6015	314	2	.	.	PUNCT
ejpam-6015	314	3	palermo	palermo	PROPN
ejpam-6015	314	4	,	,	PUNCT
ejpam-6015	314	5	22:1–74	22:1–74	NUM
ejpam-6015	314	6	,	,	PUNCT
ejpam-6015	314	7	1906	1906	NUM
ejpam-6015	314	8	.	.	PUNCT
ejpam-6015	315	1	[	[	X
ejpam-6015	315	2	3	3	X
ejpam-6015	315	3	]	]	X
ejpam-6015	315	4	s.	s.	PROPN
ejpam-6015	315	5	banach	banach	PROPN
ejpam-6015	315	6	.	.	PUNCT
ejpam-6015	316	1	sur	sur	PROPN
ejpam-6015	316	2	les	les	X
ejpam-6015	316	3	opérations	opération	NOUN
ejpam-6015	316	4	dans	dan	NOUN
ejpam-6015	316	5	les	les	X
ejpam-6015	316	6	ensembles	ensemble	NOUN
ejpam-6015	316	7	abstraits	abstrait	NOUN
ejpam-6015	316	8	et	et	PROPN
ejpam-6015	316	9	leur	leur	X
ejpam-6015	316	10	application	application	PROPN
ejpam-6015	316	11	aux	aux	PROPN
ejpam-6015	316	12	équations	équations	PROPN
ejpam-6015	316	13	intégrales	intégrale	NOUN
ejpam-6015	316	14	.	.	PUNCT
ejpam-6015	317	1	fundam	fundam	PROPN
ejpam-6015	317	2	.	.	PUNCT
ejpam-6015	317	3	math	math	PROPN
ejpam-6015	317	4	,	,	PUNCT
ejpam-6015	317	5	3:133–181	3:133–181	NUM
ejpam-6015	317	6	,	,	PUNCT
ejpam-6015	317	7	1922	1922	NUM
ejpam-6015	317	8	.	.	PUNCT
ejpam-6015	318	1	[	[	X
ejpam-6015	318	2	4	4	NUM
ejpam-6015	318	3	]	]	X
ejpam-6015	318	4	t.a.m	t.a.m	PROPN
ejpam-6015	318	5	.	.	PROPN
ejpam-6015	318	6	shatnawi	shatnawi	PROPN
ejpam-6015	318	7	w.	w.	PROPN
ejpam-6015	318	8	shatanawi	shatanawi	PROPN
ejpam-6015	318	9	.	.	PUNCT
ejpam-6015	319	1	new	new	ADJ
ejpam-6015	319	2	fixed	fix	VERB
ejpam-6015	319	3	point	point	NOUN
ejpam-6015	319	4	results	result	NOUN
ejpam-6015	319	5	in	in	ADP
ejpam-6015	319	6	controlled	control	VERB
ejpam-6015	319	7	metric	metric	ADJ
ejpam-6015	319	8	type	type	NOUN
ejpam-6015	319	9	spaces	space	NOUN
ejpam-6015	319	10	based	base	VERB
ejpam-6015	319	11	on	on	ADP
ejpam-6015	319	12	new	new	ADJ
ejpam-6015	319	13	contractive	contractive	ADJ
ejpam-6015	319	14	conditions	condition	NOUN
ejpam-6015	319	15	.	.	PUNCT
ejpam-6015	320	1	aims	aim	VERB
ejpam-6015	320	2	mathematics	mathematic	NOUN
ejpam-6015	320	3	,	,	PUNCT
ejpam-6015	320	4	8(4):9314–9330	8(4):9314–9330	NUM
ejpam-6015	320	5	,	,	PUNCT
ejpam-6015	320	6	2023	2023	NUM
ejpam-6015	320	7	.	.	PUNCT
ejpam-6015	321	1	[	[	X
ejpam-6015	321	2	5	5	X
ejpam-6015	321	3	]	]	PUNCT
ejpam-6015	321	4	w.	w.	PROPN
ejpam-6015	321	5	shatanawi	shatanawi	PROPN
ejpam-6015	321	6	a.z	a.z	PROPN
ejpam-6015	321	7	.	.	PROPN
ejpam-6015	321	8	rezazgui	rezazgui	PROPN
ejpam-6015	321	9	,	,	PUNCT
ejpam-6015	321	10	a.a	a.a	PROPN
ejpam-6015	321	11	.	.	PROPN
ejpam-6015	321	12	tallafha	tallafha	PROPN
ejpam-6015	321	13	.	.	PUNCT
ejpam-6015	322	1	common	common	ADJ
ejpam-6015	322	2	fixed	fix	VERB
ejpam-6015	322	3	point	point	NOUN
ejpam-6015	322	4	results	result	NOUN
ejpam-6015	322	5	via	via	ADP
ejpam-6015	322	6	aν	aν	NOUN
ejpam-6015	322	7	−	−	X
ejpam-6015	322	8	α−contractions	α−contraction	NOUN
ejpam-6015	322	9	with	with	ADP
ejpam-6015	322	10	a	a	DET
ejpam-6015	322	11	pair	pair	NOUN
ejpam-6015	322	12	and	and	CCONJ
ejpam-6015	322	13	two	two	NUM
ejpam-6015	322	14	pairs	pair	NOUN
ejpam-6015	322	15	of	of	ADP
ejpam-6015	322	16	self	self	NOUN
ejpam-6015	322	17	-	-	PUNCT
ejpam-6015	322	18	mappings	mapping	NOUN
ejpam-6015	322	19	in	in	ADP
ejpam-6015	322	20	the	the	DET
ejpam-6015	322	21	frame	frame	NOUN
ejpam-6015	322	22	of	of	ADP
ejpam-6015	322	23	an	an	DET
ejpam-6015	322	24	extended	extend	VERB
ejpam-6015	322	25	quasi	quasi	ADJ
ejpam-6015	322	26	b	b	NOUN
ejpam-6015	322	27	-	-	PUNCT
ejpam-6015	322	28	metric	metric	ADJ
ejpam-6015	322	29	space	space	NOUN
ejpam-6015	322	30	.	.	PUNCT
ejpam-6015	323	1	aims	aim	VERB
ejpam-6015	323	2	mathematics	mathematic	NOUN
ejpam-6015	323	3	,	,	PUNCT
ejpam-6015	323	4	8(3):7225–7241	8(3):7225–7241	NUM
ejpam-6015	323	5	,	,	PUNCT
ejpam-6015	323	6	2023	2023	NUM
ejpam-6015	323	7	.	.	PUNCT
ejpam-6015	324	1	[	[	X
ejpam-6015	324	2	6	6	NUM
ejpam-6015	324	3	]	]	PUNCT
ejpam-6015	324	4	t.	t.	PROPN
ejpam-6015	324	5	abdeljawad	abdeljawad	PROPN
ejpam-6015	324	6	m.	m.	PROPN
ejpam-6015	324	7	joshi	joshi	PROPN
ejpam-6015	324	8	,	,	PUNCT
ejpam-6015	324	9	a.	a.	PROPN
ejpam-6015	324	10	tomar	tomar	PROPN
ejpam-6015	324	11	.	.	PUNCT
ejpam-6015	325	1	on	on	ADP
ejpam-6015	325	2	fixed	fix	VERB
ejpam-6015	325	3	points	point	NOUN
ejpam-6015	325	4	,	,	PUNCT
ejpam-6015	325	5	their	their	PRON
ejpam-6015	325	6	geometry	geometry	NOUN
ejpam-6015	325	7	and	and	CCONJ
ejpam-6015	325	8	application	application	NOUN
ejpam-6015	325	9	to	to	ADP
ejpam-6015	325	10	satellite	satellite	NOUN
ejpam-6015	325	11	web	web	NOUN
ejpam-6015	325	12	coupling	coupling	NOUN
ejpam-6015	325	13	problem	problem	NOUN
ejpam-6015	325	14	in	in	ADP
ejpam-6015	325	15	s−metric	s−metric	ADJ
ejpam-6015	325	16	spaces	space	NOUN
ejpam-6015	325	17	.	.	PUNCT
ejpam-6015	326	1	aims	aim	VERB
ejpam-6015	326	2	mathematics	mathematic	NOUN
ejpam-6015	326	3	,	,	PUNCT
ejpam-6015	326	4	8(2):4407	8(2):4407	NUM
ejpam-6015	326	5	–	–	PUNCT
ejpam-6015	326	6	4441	4441	NUM
ejpam-6015	326	7	,	,	PUNCT
ejpam-6015	326	8	2023	2023	NUM
ejpam-6015	326	9	.	.	PUNCT
ejpam-6015	327	1	[	[	X
ejpam-6015	327	2	7	7	NUM
ejpam-6015	327	3	]	]	X
ejpam-6015	327	4	m.c	m.c	PROPN
ejpam-6015	327	5	.	.	PROPN
ejpam-6015	327	6	reurings	reurings	PROPN
ejpam-6015	327	7	a.c.m	a.c.m	PROPN
ejpam-6015	327	8	.	.	PROPN
ejpam-6015	327	9	ran	run	VERB
ejpam-6015	327	10	.	.	PUNCT
ejpam-6015	328	1	a	a	DET
ejpam-6015	328	2	fixed	fix	VERB
ejpam-6015	328	3	point	point	NOUN
ejpam-6015	328	4	theorem	theorem	VERB
ejpam-6015	328	5	in	in	ADP
ejpam-6015	328	6	partially	partially	ADV
ejpam-6015	328	7	ordered	order	VERB
ejpam-6015	328	8	sets	set	NOUN
ejpam-6015	328	9	and	and	CCONJ
ejpam-6015	328	10	some	some	DET
ejpam-6015	328	11	applications	application	NOUN
ejpam-6015	328	12	to	to	PART
ejpam-6015	328	13	matrix	matrix	VERB
ejpam-6015	328	14	equations	equation	NOUN
ejpam-6015	328	15	.	.	PUNCT
ejpam-6015	329	1	proc	proc	PROPN
ejpam-6015	329	2	.	.	PUNCT
ejpam-6015	330	1	amer	amer	PROPN
ejpam-6015	330	2	.	.	PUNCT
ejpam-6015	330	3	math	math	PROPN
ejpam-6015	330	4	.	.	PUNCT
ejpam-6015	331	1	soc	soc	PROPN
ejpam-6015	331	2	,	,	PUNCT
ejpam-6015	331	3	132:1435–1443	132:1435–1443	NUM
ejpam-6015	331	4	,	,	PUNCT
ejpam-6015	331	5	2004	2004	NUM
ejpam-6015	331	6	.	.	PUNCT
ejpam-6015	332	1	[	[	X
ejpam-6015	332	2	8	8	NUM
ejpam-6015	332	3	]	]	X
ejpam-6015	332	4	p.	p.	PROPN
ejpam-6015	332	5	vetro	vetro	PROPN
ejpam-6015	332	6	b.	b.	PROPN
ejpam-6015	332	7	samet	samet	PROPN
ejpam-6015	332	8	,	,	PUNCT
ejpam-6015	332	9	c.	c.	PROPN
ejpam-6015	332	10	vetro	vetro	PROPN
ejpam-6015	332	11	.	.	PUNCT
ejpam-6015	333	1	fixed	fix	VERB
ejpam-6015	333	2	point	point	NOUN
ejpam-6015	333	3	theorems	theorem	VERB
ejpam-6015	333	4	for	for	ADP
ejpam-6015	333	5	α−	α−	ADP
ejpam-6015	333	6	ψ	ψ	NOUN
ejpam-6015	333	7	-	-	ADJ
ejpam-6015	333	8	contractive	contractive	ADJ
ejpam-6015	333	9	type	type	NOUN
ejpam-6015	333	10	mappings	mapping	NOUN
ejpam-6015	333	11	.	.	PUNCT
ejpam-6015	334	1	nonlinear	nonlinear	ADJ
ejpam-6015	334	2	anal	anal	PROPN
ejpam-6015	334	3	,	,	PUNCT
ejpam-6015	334	4	75:2154–2165	75:2154–2165	NUM
ejpam-6015	334	5	,	,	PUNCT
ejpam-6015	334	6	2012	2012	NUM
ejpam-6015	334	7	.	.	PUNCT
ejpam-6015	335	1	[	[	X
ejpam-6015	335	2	9	9	NUM
ejpam-6015	335	3	]	]	X
ejpam-6015	335	4	e.	e.	PROPN
ejpam-6015	335	5	karapinar	karapinar	PROPN
ejpam-6015	335	6	s.	s.	PROPN
ejpam-6015	335	7	sahmim	sahmim	PROPN
ejpam-6015	335	8	h.	h.	PROPN
ejpam-6015	335	9	aydi	aydi	PROPN
ejpam-6015	335	10	,	,	PUNCT
ejpam-6015	335	11	a.	a.	PROPN
ejpam-6015	335	12	felhi	felhi	PROPN
ejpam-6015	335	13	.	.	PUNCT
ejpam-6015	336	1	a	a	DET
ejpam-6015	336	2	nadler	nadler	NOUN
ejpam-6015	336	3	-	-	PUNCT
ejpam-6015	336	4	type	type	NOUN
ejpam-6015	336	5	fixed	fix	VERB
ejpam-6015	336	6	point	point	NOUN
ejpam-6015	336	7	theorem	theorem	VERB
ejpam-6015	336	8	in	in	ADP
ejpam-6015	336	9	dislocated	dislocate	VERB
ejpam-6015	336	10	spaces	space	NOUN
ejpam-6015	336	11	and	and	CCONJ
ejpam-6015	336	12	applications	application	NOUN
ejpam-6015	336	13	.	.	PUNCT
ejpam-6015	337	1	miscolc	miscolc	PROPN
ejpam-6015	337	2	math	math	PROPN
ejpam-6015	337	3	.	.	PUNCT
ejpam-6015	338	1	notes	note	NOUN
ejpam-6015	338	2	,	,	PUNCT
ejpam-6015	338	3	19(1):111–124	19(1):111–124	NUM
ejpam-6015	338	4	,	,	PUNCT
ejpam-6015	338	5	2018	2018	NUM
ejpam-6015	338	6	.	.	PUNCT
ejpam-6015	339	1	[	[	X
ejpam-6015	339	2	10	10	NUM
ejpam-6015	339	3	]	]	X
ejpam-6015	339	4	s.b	s.b	PROPN
ejpam-6015	339	5	.	.	PROPN
ejpam-6015	339	6	nadler	nadler	PROPN
ejpam-6015	339	7	.	.	PUNCT
ejpam-6015	339	8	multivalued	multivalue	VERB
ejpam-6015	339	9	contraction	contraction	NOUN
ejpam-6015	339	10	mappings	mapping	NOUN
ejpam-6015	339	11	.	.	PUNCT
ejpam-6015	340	1	pac	pac	PROPN
ejpam-6015	340	2	.	.	PUNCT
ejpam-6015	341	1	j.	j.	PROPN
ejpam-6015	341	2	math	math	PROPN
ejpam-6015	341	3	,	,	PUNCT
ejpam-6015	341	4	30:475–488	30:475–488	NUM
ejpam-6015	341	5	,	,	PUNCT
ejpam-6015	341	6	1969	1969	NUM
ejpam-6015	341	7	.	.	PUNCT
ejpam-6015	342	1	[	[	X
ejpam-6015	342	2	11	11	NUM
ejpam-6015	342	3	]	]	X
ejpam-6015	342	4	n.	n.	PROPN
ejpam-6015	342	5	shahzad	shahzad	PROPN
ejpam-6015	343	1	j.h	j.h	PROPN
ejpam-6015	343	2	.	.	PROPN
ejpam-6015	343	3	asl	asl	PROPN
ejpam-6015	343	4	,	,	PUNCT
ejpam-6015	343	5	s.	s.	PROPN
ejpam-6015	343	6	rezapour	rezapour	VERB
ejpam-6015	343	7	.	.	PUNCT
ejpam-6015	344	1	on	on	ADP
ejpam-6015	344	2	fixed	fix	VERB
ejpam-6015	344	3	points	point	NOUN
ejpam-6015	344	4	of	of	ADP
ejpam-6015	344	5	α−ψ	α−ψ	NOUN
ejpam-6015	344	6	-	-	PUNCT
ejpam-6015	344	7	contractive	contractive	ADJ
ejpam-6015	344	8	multifunctions	multifunction	NOUN
ejpam-6015	344	9	.	.	PUNCT
ejpam-6015	345	1	fixed	fix	VERB
ejpam-6015	345	2	point	point	NOUN
ejpam-6015	345	3	theory	theory	NOUN
ejpam-6015	345	4	appl	appl	NOUN
ejpam-6015	345	5	,	,	PUNCT
ejpam-6015	345	6	2012:2012	2012:2012	NUM
ejpam-6015	345	7	,	,	PUNCT
ejpam-6015	345	8	2012	2012	NUM
ejpam-6015	345	9	.	.	PUNCT
ejpam-6015	346	1	[	[	X
ejpam-6015	346	2	12	12	NUM
ejpam-6015	346	3	]	]	PUNCT
ejpam-6015	346	4	j.	j.	PROPN
ejpam-6015	346	5	siegel	siegel	PROPN
ejpam-6015	346	6	j.p	j.p	PROPN
ejpam-6015	346	7	.	.	PROPN
ejpam-6015	346	8	aubin	aubin	PROPN
ejpam-6015	346	9	.	.	PUNCT
ejpam-6015	347	1	fixed	fix	VERB
ejpam-6015	347	2	points	point	NOUN
ejpam-6015	347	3	and	and	CCONJ
ejpam-6015	347	4	stationary	stationary	ADJ
ejpam-6015	347	5	points	point	NOUN
ejpam-6015	347	6	of	of	ADP
ejpam-6015	347	7	dissipative	dissipative	ADJ
ejpam-6015	347	8	multivalued	multivalue	VERB
ejpam-6015	347	9	maps	map	NOUN
ejpam-6015	347	10	.	.	PUNCT
ejpam-6015	348	1	proceedings	proceeding	NOUN
ejpam-6015	348	2	of	of	ADP
ejpam-6015	348	3	the	the	DET
ejpam-6015	348	4	american	american	PROPN
ejpam-6015	348	5	mathematical	mathematical	PROPN
ejpam-6015	348	6	society	society	NOUN
ejpam-6015	348	7	,	,	PUNCT
ejpam-6015	348	8	78(3):391–398	78(3):391–398	PROPN
ejpam-6015	348	9	,	,	PUNCT
ejpam-6015	348	10	1980	1980	NUM
ejpam-6015	348	11	.	.	PUNCT
ejpam-6015	349	1	h.	h.	PROPN
ejpam-6015	349	2	aydi	aydi	VERB
ejpam-6015	349	3	et	et	PROPN
ejpam-6015	349	4	al	al	PROPN
ejpam-6015	349	5	.	.	PUNCT
ejpam-6015	349	6	/	/	SYM
ejpam-6015	349	7	eur	eur	PROPN
ejpam-6015	349	8	.	.	PUNCT
ejpam-6015	350	1	j.	j.	PROPN
ejpam-6015	350	2	pure	pure	PROPN
ejpam-6015	350	3	appl	appl	PROPN
ejpam-6015	350	4	.	.	PROPN
ejpam-6015	350	5	math	math	PROPN
ejpam-6015	350	6	,	,	PUNCT
ejpam-6015	350	7	18	18	NUM
ejpam-6015	350	8	(	(	PUNCT
ejpam-6015	350	9	2	2	NUM
ejpam-6015	350	10	)	)	PUNCT
ejpam-6015	350	11	(	(	PUNCT
ejpam-6015	350	12	2025	2025	NUM
ejpam-6015	350	13	)	)	PUNCT
ejpam-6015	350	14	,	,	PUNCT
ejpam-6015	350	15	6015	6015	NUM
ejpam-6015	350	16	14	14	NUM
ejpam-6015	350	17	of	of	ADP
ejpam-6015	350	18	14	14	NUM
ejpam-6015	351	1	[	[	SYM
ejpam-6015	351	2	13	13	NUM
ejpam-6015	351	3	]	]	X
ejpam-6015	351	4	l.v	l.v	NOUN
ejpam-6015	351	5	.	.	PROPN
ejpam-6015	351	6	hot	hot	PROPN
ejpam-6015	351	7	.	.	PUNCT
ejpam-6015	352	1	fixed	fix	VERB
ejpam-6015	352	2	point	point	NOUN
ejpam-6015	352	3	theorems	theorem	NOUN
ejpam-6015	352	4	for	for	ADP
ejpam-6015	352	5	multivalued	multivalue	VERB
ejpam-6015	352	6	mapping	mapping	NOUN
ejpam-6015	352	7	.	.	PUNCT
ejpam-6015	353	1	commentationes	commentatione	NOUN
ejpam-6015	353	2	mathematicae	mathematicae	VERB
ejpam-6015	353	3	universitatis	universitatis	PROPN
ejpam-6015	353	4	carolinae	carolinae	PROPN
ejpam-6015	353	5	,	,	PUNCT
ejpam-6015	353	6	23:137–145	23:137–145	PROPN
ejpam-6015	353	7	,	,	PUNCT
ejpam-6015	353	8	1982	1982	NUM
ejpam-6015	353	9	.	.	PUNCT
ejpam-6015	354	1	[	[	X
ejpam-6015	354	2	14	14	NUM
ejpam-6015	354	3	]	]	X
ejpam-6015	354	4	f.	f.	PROPN
ejpam-6015	354	5	khojasteh	khojasteh	PROPN
ejpam-6015	354	6	w.s	w.s	PROPN
ejpam-6015	354	7	.	.	PROPN
ejpam-6015	354	8	du	du	PROPN
ejpam-6015	354	9	.	.	PUNCT
ejpam-6015	355	1	new	new	ADJ
ejpam-6015	355	2	results	result	NOUN
ejpam-6015	355	3	and	and	CCONJ
ejpam-6015	355	4	generalizations	generalization	NOUN
ejpam-6015	355	5	for	for	ADP
ejpam-6015	355	6	approximate	approximate	ADJ
ejpam-6015	355	7	fixed	fix	VERB
ejpam-6015	355	8	point	point	NOUN
ejpam-6015	355	9	property	property	NOUN
ejpam-6015	355	10	and	and	CCONJ
ejpam-6015	355	11	their	their	PRON
ejpam-6015	355	12	applications	application	NOUN
ejpam-6015	355	13	.	.	PUNCT
ejpam-6015	356	1	abstr	abstr	PROPN
ejpam-6015	356	2	.	.	PUNCT
ejpam-6015	356	3	appl	appl	PROPN
ejpam-6015	356	4	.	.	PUNCT
ejpam-6015	357	1	anal	anal	PROPN
ejpam-6015	357	2	,	,	PUNCT
ejpam-6015	357	3	1:581267	1:581267	NUM
ejpam-6015	357	4	,	,	PUNCT
ejpam-6015	357	5	2014	2014	NUM
ejpam-6015	357	6	.	.	PUNCT
ejpam-6015	358	1	[	[	X
ejpam-6015	358	2	15	15	NUM
ejpam-6015	358	3	]	]	X
ejpam-6015	358	4	m.a	m.a	PROPN
ejpam-6015	358	5	.	.	PROPN
ejpam-6015	358	6	kutbi	kutbi	PROPN
ejpam-6015	358	7	n.	n.	PROPN
ejpam-6015	358	8	hussain	hussain	PROPN
ejpam-6015	358	9	,	,	PUNCT
ejpam-6015	358	10	i.	i.	PROPN
ejpam-6015	358	11	iqbal	iqbal	PROPN
ejpam-6015	358	12	.	.	PUNCT
ejpam-6015	359	1	fixed	fix	VERB
ejpam-6015	359	2	point	point	NOUN
ejpam-6015	359	3	theorems	theorem	NOUN
ejpam-6015	359	4	for	for	ADP
ejpam-6015	359	5	manageable	manageable	ADJ
ejpam-6015	359	6	contractions	contraction	NOUN
ejpam-6015	359	7	with	with	ADP
ejpam-6015	359	8	application	application	NOUN
ejpam-6015	359	9	to	to	ADP
ejpam-6015	359	10	integral	integral	ADJ
ejpam-6015	359	11	equations	equation	NOUN
ejpam-6015	359	12	.	.	PUNCT
ejpam-6015	360	1	journal	journal	NOUN
ejpam-6015	360	2	of	of	ADP
ejpam-6015	360	3	function	function	NOUN
ejpam-6015	360	4	spaces	space	NOUN
ejpam-6015	360	5	,	,	PUNCT
ejpam-6015	360	6	2017:10	2017:10	ADJ
ejpam-6015	360	7	pages	page	NOUN
ejpam-6015	360	8	,	,	PUNCT
ejpam-6015	360	9	2017	2017	NUM
ejpam-6015	360	10	.	.	PUNCT
ejpam-6015	361	1	[	[	X
ejpam-6015	361	2	16	16	NUM
ejpam-6015	361	3	]	]	X
ejpam-6015	361	4	h.	h.	PROPN
ejpam-6015	361	5	aydi	aydi	VERB
ejpam-6015	361	6	a.	a.	PROPN
ejpam-6015	361	7	felhi	felhi	PROPN
ejpam-6015	361	8	.	.	PUNCT
ejpam-6015	362	1	new	new	ADJ
ejpam-6015	362	2	fixed	fix	VERB
ejpam-6015	362	3	point	point	NOUN
ejpam-6015	362	4	results	result	NOUN
ejpam-6015	362	5	for	for	ADP
ejpam-6015	362	6	mult	mult	PROPN
ejpam-6015	362	7	-	-	PUNCT
ejpam-6015	362	8	valued	value	VERB
ejpam-6015	362	9	maps	map	NOUN
ejpam-6015	362	10	via	via	ADP
ejpam-6015	362	11	manageable	manageable	ADJ
ejpam-6015	362	12	functions	function	NOUN
ejpam-6015	362	13	and	and	CCONJ
ejpam-6015	362	14	an	an	DET
ejpam-6015	362	15	application	application	NOUN
ejpam-6015	362	16	on	on	ADP
ejpam-6015	362	17	a	a	DET
ejpam-6015	362	18	boundary	boundary	ADJ
ejpam-6015	362	19	value	value	NOUN
ejpam-6015	362	20	problem	problem	NOUN
ejpam-6015	362	21	.	.	PUNCT
ejpam-6015	363	1	u.p.b	u.p.b	PROPN
ejpam-6015	363	2	.	.	PUNCT
ejpam-6015	364	1	sci	sci	PROPN
ejpam-6015	364	2	.	.	PUNCT
ejpam-6015	364	3	bull	bull	PROPN
ejpam-6015	364	4	.	.	PUNCT
ejpam-6015	364	5	,	,	PUNCT
ejpam-6015	364	6	series	series	PROPN
ejpam-6015	364	7	a	a	PRON
ejpam-6015	364	8	,	,	PUNCT
ejpam-6015	364	9	80(1):1–12	80(1):1–12	NUM
ejpam-6015	364	10	,	,	PUNCT
ejpam-6015	364	11	2018	2018	NUM
ejpam-6015	364	12	.	.	PUNCT